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All]USAAML TECHNICAL REPORT 65-20
"let CURVED JET FLOWS
VOLUME I
•' •. LJUE._• r ,"tsi•.-.. '.4,,"RI I
I.. ARMY AVIATIOI MATERIEL LAIIIAIIIIES
FlRT EUSTIS, VIIIINIA
CONTRACT DA 44-177-AMC-238(T)PETER R. PAYNE, INC.
Vt
I".
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Disclaimer
The findings in this report are not to be construed as an official Departmentof the Army position, unless so designated by other authorized documents.
When Government drawings, specifications, or other data are used for anypurpose other than in connection with a definitely related Government pro-curement operation, the United States Government thereby incurs no re-sponsibility nor any obligation whatsoever; and the fact that the Governmentmay have formulated, furnished, or in any way supplied the said drawings,specifications, or other data is not to be regarded by implication or other-wise as in any manner licensing the holder or any other person or corpora-tion, or conveying any rights or permission, to manufacture, use or sellany patented invention that may in any way be related thereto.
Dismaltion Instructions
Destroy this report when it is no longr neded. Do not return it to theoriginator.
U S ARMY TRANSPORTATION RIESEARCH COMMANDPOWU VJ SNS 83504 4
The present study was undertaken to describe more accurately viscous
influences on the annu±ar jet floL The theory developed also
correlated with inlet flow, Coanda flow, and jet-flap experimintal
data.
The report has been reviewed by the U.S. Arm Transportation Research
Co and, and the data contained in it are considered to be valid.
The report is published for the dissemination of information.
r4
NOTE
On I March 1965, after MU rpo•t hadbeen prspamed. the nm of this commeandwas hanged from U.S. Army TransportationResearch Commnad tot
U.S. AMY. AVIAION NA,4T.EL LABORATOR,•S
Task 1D12140:1.A14203Contract DA 44-177-AMC-238(T)
USAAML Technical Report 65-ZOMay 1965
CURVED JET FLOWS
PnP. by•is1a Paikiwm Dinw
Awkwill, VIr7IamI 2"52
for
U. S. ARMY AVIATION MATERIEL LABORATORIESFORT EUSTU, V=lDGZA
ABSTRACT
A simple equation is derived to describe curvilinear flow, and this is then ap-plied to various practical problems. In the case of annular jet flow, previoustheories are shown to be approximate solutions to the general equation; themore exact solution of this report is shown to give better agreement with ex-periment.
The new theory is also applied to the flow of air in a curved duct, the flow intoan Intake, the jet flap, and Coanda flow. Comparison with experiment againgives good agreement.
Because curvilinear flow implies diffusion, the theory of diffusion is studiedin some c'tail, and a general theory is developed for the total head lost in arapid di.-:sion. When applied to the diffusion loss measured in the nozzlz ofan anm ar jet, and the analogous losses in a curved duct, the theory gives ex-oellent Agreement with experiment.
The various investigations oover fairly wide ares in subsonic aerodynamics.Thus, i: has proved tmpossile to work out the appdicatoons of the theory com-pletely for all the case ooamsldsed. The same is true of the experimental workreported, and fther work ft both aras is rggested.
iV
PREFACE
The work herein reported was carried out at Peter R. Payne, Inc., Rockville,Maryland, in compliance with U. S. Army Transportation Research CommandContract DA 44-177-AMC-238(T).
The principal investigator was Mr. Peter R. Payne, and the Project Engineerwas Mr. Alastair Anthony. In general, Mr. Payne was responsible for thetheoretical work and the basic concepts introduced, and Mr. Anthony for theexperimental work and the program management.
Mr. Anthony was ably assisted by Messrs. David Ullman and James Ozenhamof the Payne, Inc. staff, and by Payne's associate consultant, Professor GeorgePick.
The technical adminic-tration representative of the U. L. Army TransportationResearch Command was Mr. William E. ickles, whose invaluable assistanoeis gratefully acknowledged.
I V'a
CONTENTS
ABSTRACT ......................... o................ fit
PREFACE ........... .. .... . .................. v
LIST OF ILLUSTRATIONS ........ .......... ... .. . .. . xvii
LIST OF SYMBOLS ..................... ......... .. . xxvii
CHAPTER ONE - SUMMARY AND DISCUSSION .1
THE FUNDAMENTAL PROBLEM ...................... 1
DIFFUSION LOSSES ........... ............... 5
ANNULAR JET THEORY ................... ........ 8
CHAPLIN'S SOLUTION (THIN-JET THEORY) ............. 9
THE CROSS-SrANTON-JONES SOLUTION(EXPONENTIAL THEORY) ........ ......... .... ..... 11
PINNES' BOLUTION (FREE-VORTEX FLOW) ....... ... 12
PAYNE'8 CORRECTION FOR JET CnVATUM ............ 14
THE ANNULAR JET SOLUTIOII OF 7= REPORT ......... 14
MOMENTUM BALANCE OF THS ANIWLAR JET .........o 19
COANDA JET THEORY............ ............. 20
INTAKE FLOW THEORY............ .a............ o24
MOMENTUM DRAG .... o ...... a a a o o .......... 26
THE JET FLAP. #. o . .... eec.. o s o o o o @ o o * o..26
JET ISSUING NORMAL TO A FREEZ-rRhAM FLOW ..... . o . .31
S Vi
CHAPTER TWO - GENERAL DIFFUSION THEORY .............. 35
THE FLOW OF A JET THROUGH A DISCONTINUITY INSTATIC PRESSURE ................................ 35
SOLUTION FOR CONSeRVATION OF TOTAL HEAD ......... .35
THE GENERAL MOMENTUM SOLUTION ................ 36
BORDA-CARNOT DIFFUSION IN TWO-DIMENSIONAL FLOW. . Al
VALUES OF THE SHAPE FACTORS A AND 4 ........ .48
THE LIMITING VELOCITY CHANGE ................... £50
CHAPTER THREE - THE EQUATION FOR CURVILINEAR TWO-DIMENSIONAL FLOW ................................... 58
THE EQUATION OF MOTION ........................ .58
A LINEAR TRANSFORMATION TO JET ORDIATES ....... £0
NONLINEAR TANFORMATIONS ................... S1
CHAPTER FOUR - DEYCI FL4W ANNULAR JET THEORY FORCONSITANT TOTAL R .... . .0.. . . .. ... ........ .£3
GENERAL MOMENTUM BALANCE CONSIERATIONS ...... £3
A Th.k Jet bhian (PLso' Goeometry) ........... £3
A ickJ et b abtto. . .o e ................ e.*..... A7
The Variation of Curvature Across a Thick Jet ...... S7
The Height Parameter ......... ......... .... .. 69
Three-Dimensional Effects .................... .70
vifi
SOME GENERAL RELATIONS FOR THE FLOW PARAMETERS
OF AN ANNULAR JET ............................... 71
The Local Jet Velocity ............................. 72
The Jet Mass Flow ...................................... 72
The Nozzle Momentum Flux .................... 72
Total Nozzle Force .......... ................... 72
Jet Power ................................ 73
Total Lift .............. . 0..... ............ 73
SOLUTION OF THE CURVED JET EQUATION FOR CONSTANTTOTAL PRESSURE. .e..... * * %e *. eea......... . .. 73
Local Jet Velocity .......................... 74
Jet Maos Flow ........................... ... 74
The Nozzle Momeatm Flu.. . . . .a 9 .. e .. . . .a. . . . 74
Totdl Jet Fror* ... .0 *........ ..... 74
The Jet Oarvafre PUmeter..,,,,,,. ....... .. 75
MTZ .Dlk e P N .12 ,,,,,,, ................ 76
OtherRults. ,,....,,,t......,..........80
THE CUinON PRuin 2PA IRAM3TER.,,. ....... , ... .. 87
The General Ezpmirsou fokit 6 ...... ... *..... ..
The Constant Total Head Solution ................ 93
The Solution for Free-Vortex Flow (• 1.0) ....... 94
ix
The Solution for Exponential Theory Flow ( *7 = 0). . .94
GENERAL 9OLUTIONS FOR FREE-VORTEX FLOW WITHCONSTANT TOTAL PRF MRE ...................... 95
Calhulation of Mass Flow ...................... 95
Calculation to the Local Velocity ..... ........... 95
The Nozzle Mometum Flux ..................... 96
The Total Jet Force ......................... 96
The Jet Power ................. ................. 96
Momentum Balanoe .......................... 96
GENERAL SOLUTIt5W POR CONSTANT PADRIU FLOW(EXPONENTIAL THEORY) WITH CONSTANT TOTALPRESSURE .... .. .... •. • ... ...... ............... 97
The LJooalJMt Velocity ........ ................... 98
The JehIMassFlow... . .. ...... .. ...... .. .... 98
Tbe Jet ie No w* 0.. . .. . .. 0 .. .0 .. . . . . ....... . 98
lTe Nlag Ml'oNOe...................0..98
30HZ WZUMTAL RZWLTS AT CONNTANT TOTALpi w aU.3 . . . . . .e a 0 a 0 0 & a a. S * *.. .... . *.. . . .. 100
Tbe Anla Jet Te"t Rig . . o . .. . ... ........ .. . .100
Thel ZperimenG. ................ .... .. .. .. 100
In tm tation . a . . .. .4. .0. . . . . . . .0 .100
x
Flow at the Throat .......................... 103
Flow at the Nozzle ................. ......... 103
Flow Adjacent to the Annulas r Jet ................. 110
CHAPTER FIVE - THE EFFECT OF TOTAL PRESISTE VARIATIONAC(ROSS AN ANNULAR JET .............................. 116
THE SPECIAL CASE OF CONSTANT JET VELOCITY ........ 116
Determination of the Jet Curvature Pa!m.metev ...... 118
Calculation of +,,For Free-Vortex I low ......... 118
Calculation of T. for Constan+ Radiua Flow ........ 110
Discussion of Results ....... ................. 119
THE INFLUENCE OF THE TOTAL PRESSJRE DETEBUTIONON CUSHION PRESSJRE IN rERNS OF MEAN TOTAL PRESSURE,FOR FREE-VORTEX FLOW ...................... * * .122
The Jet Thickness Anomaly .................... 126
A Gemeral Llinsa Varatiom.................... 131
A General Pow Iaw VW0tka........... ..... 13
CUSMoN iPti AS A MTI0 OF NUtN TOTAL pMWERXWiT A LE&A VARKTW IN TOTAL PMn ACWUW THEJET (EtPOKENTAL THIBOG) . .*. . .*. . . . . . . . . . . . . . . .a 134
THE EFFECT OF A LWIER TOTAL qD VAMRITION OF THECU IUUON PRERZ PAAMETER A ....... 137
Variation in Cuswion Presaure Parameter Acoording toEzponential Flow Theory With a LInear Gradient of TotalPressure Across the Jet . ........... a 0 . . a 0 . a .139
xi
Variation in Cushion Pressure Parameter Accordingto Free-Vortex Flow With a Linear Gradient of TotalPressure Across the Jet ...................... 141
Discussion of the Results ...... . . ............ 144
CHAPTER SIX - VISCOUS MIXING EFFECTS IN THE ANNULAR JET. .147
THE "AIR FRICTION" CONCEPT ..................... 154
A GENERAL THEORY OF MIXING LOSS ................. 162
THE STATIC PRESSURE IN THE PRIMARY VORTEX. ....... 174
CHAPTER SEVEN - A SMALL PERTURBATION THEORY OF DIFFUSIONLOSSES IN AN ANNULAR JET NOZZLE ...................... 176
APPROXIMATE SOLUTION FOR A "STRAIGHT" NOZZLE .... 179
MEAN TOTAL PRESSURE LOSS ...................... 180
TOTAL POWER LOW DUE TO DIFFUKON. .............. 182
CHAPTER EIGHT - CCAM JET FLOW ..................... 185
SOLTI1ON FOR COWNTANT TOTAL HEAD..... .......@ a 186
APOUM&TR SOLUTIO FOR THE BOUNDARY LAYER
CHAPTER MWE - TWO-4UUKAL FLOW IN A CURVED DUCT
WITH COUIUIAII' TOTAL BRAD............... .. .0. 0 .. .a 195
$?ATIA PROSURE AND VELOCITY DITMUTION ........ 196
THE CONSTANT VELOCITY STREAMLINE .............. 198
MAN FLOW DWfSUTION AROUND THE BEND......... .200
TOTAL HEAD LOW ON THE OUTSIDE OF THE CONSTANTVELOCTY SrRAMLINE .. . ....... ... 0. ........... 200
xii
TOTAL HEAD LOS ON THE INSIDE OF THE CONSTANTVELOCITY STREAMLINE .......................... 203
MEAN TOTAL PRESSURE LOSS ACROSS THE DUCT ........ 204
THE LOSS INCREMENT DUE TO SKIN FRICTION .......... 204
COMPARISON WITH EXPERIMENT .................... 206
*kPTER TEN - FLOW INTO A TWO-DIMENSIONAL FLUSH INTAKE .208
MOMENTUM EQUATIONS .......................... 210
THE FLOW EQUATION FOR CONSTANT TOTAL HEAD ...... 211
INTAKE MASS FLOW .............. 212
THE FLOW CURVATURE PARAMETER 7 FOR CONSERVATIDNOF MOMENTUM ............................... • o213
THE STATIC PRESSURE DSTRUTION A#.......e ..... .214
LIMIT SOLUTIONS FOR 0, =WANT..... 216
SOLUTIONS FOR7 1. 65 a WWrAXT ...........217
TUE FLOW XWATION WrE £1 UMI!AM AA DA SA
LAYER ................................. .... 219
SOLUTION F(t A T= UfT91801 SWIAXY LAYTIM..... = 1
UoMI 32x3NNEITAL M3Amn33yS............... 23
Intak Pom Ima/ems................... .... u
Ezpsrbnw 09 b9a MeaduWsmZ o li Powr EIWmoy.. 22s
Calibratioa of Yaw Prbo, for Static PrummMesrenm t ............. .................
xiu
Comparison With Theoretical Values ......... s . a229
Comparison with Zero Tunnel Speed Case ........... 237
CHAPTER ELEVEN - SOME GENERAL PROBLEMS OF A JET DIS-CHARGING ACROSS A FREE-STREAM FLOW .................. 240
THRUST LOSS DUE TO MDIING ...................... 240
SOURCES OF APPARENT JET DRAG COMMON TO ALL JETCONFIGURATIONS ............................... 243
The Effect of Pressure Due to Free-Stream Flowv ..... 244
The Effect of "Boundary Layer Pumping"............ 247
Jet-Induced Turbulence ..... .......................... 248
Nozzle DifW sion Losses ...................... 249
A SIMPLE HYPOTHEMS FOR JET PRESURE IN A FREE-STREAM FLOW.. . . * % * s .. .e. . . . * . e. .. . . * . *. . . .. ... . 250
KZ EXPZISKMlAL OUBVATVMION................ .24
Flmws m ,mre ft Pi 9boond ................ 254
Veassla In f hr Isoand ............ . ..... 254
11DWYM MI o • woeK .................. 265
Amoder. 1'hoo9M .......... ...... .. ...... 265R3O~m3 mM b.. 18 5Tbm .... * a o *, . e . o # o. * o e * e 266
er d Daet Flow T ory.... .. .. .. ... .... . ..
Coema Flow .. .... ..... .. . ......... 267
Diffniouu Theoy . ..... .... . . e. . . #.... 267
Jet Flap Theory. ......................... .27
xiv
BEBLIOGRAPHY .................................... .268
APPENDIX I- SCALE EFFECT IN AINULAR JET FLOW o......... 273
APPENDIX II- ESTIMATION OF DRAG FROM TOTAL HEADSURVEYS ... o o~~o....... o es... *........ . .. . .. . . ...... 277
APPENDIX III - A NOTE ON THE THEORY OF A FREE VORTEX .... 278
APPENDIX IV - DIFFUSION LOSS MEASUREMENTS IN A PRE-CURVED NOZZLE .............................. .. . . . .281
xv
LIST OF ILLUSTRATION'S
1 Curveca Flow Fields Considered in This Report. .... 2
2 The "Thrust Hypothesis" for a Jet Flap in Inviscid Flow . 3
3 Centrifugal Force on a Curved Jet Element. . . . . . 5
4 Borda-Carnot Diffusion ....... ............ 6
5 Chaplin's Curved Jet Geometry ............ 10
6 Physical Interpretation of the Constant Radius ofCurvature Jet . . . . . . . . . . . . . . 11
7 Pinnes' "Free Vortex" Assumption for Annular Jet Flo .- 13
8 Effect of Ground Clearance on Jet Radius of Curvature . . 15
9 Comparison of the New Invisoid Flow Theory WithPrevious Theories. . . . . . . . . . . . .17
10 Comparis of the New Iviscid Flow Theory WithSome Zpsrimental DN(@ ( - . . . . . . . . is
11 Momemtum Balano. Geometry. . . . . . . . . . 19
12 Kqaivalese of A=mlw Jhi &d Oomd Jet rlows. . . . 20
13 Variation A 111 htUO # With the MixingPrSehsent &A. . . .*. . . . . . . 21
14 Variation of Aumeatatlom ' Ml. With Ditalan Wficieny,Assuming Optimum Muixg Pressure. . . . . . . . 22
15 Variation of Entrainment Ratio With Jet Thiokmess for aRight-Angle CoandlBsmd . . . . . . . . . . . 23
16 A Coanda Protile Designed for Maximum ThrustAugmentation .......... . .. . 24
Xvii
17 Leading Edge Separation With Intake Flow . . . . . . 25
18 Twenty-Percent Elliptical Aerofoil in the SmokeTunnel. No Air Supply to Jet Flap . . . . . . . . 27
19 Streamlines Around the Elliptical Aerofoil When theJet Flap is Operating . ..... ............ 28
20 Jet Flap Experiment in the Two-Dimensional WindTunnel. . . . . ....... ................ 29
21 Results of Total Head Traverses 3.25 Chords Behindthe Jet Flapped Aerofoil, With the Jet Flap On and Off . . 30
22 Observed Flow Patterns, With and Without Jet FlapOperating . . ...... ..................... 32
23 Assumed Geometry. . . . . . . . . . . . .33
24 Geometry of Flow Through a Pressure Change. . . . . 35
25 Change In Momenoum Flux Through a Static PressureChamp for Oe-Dime /mtomul Flow . . . . . . . . 38
26 Cout d sd U tmatmd Flow Throug aatio Prwe Cqp . . . . . . . . . . . . 39
27 Thewt• uhr9M d4 . . .h ...... ... 40
26 Two Jd 1im OswAjms . . . . . . . . . . . 40
29 A buss iet Zomwks . . . . . ... .. . . . 41
30 Vara'Wb ua a l&usm DlMsion . . . . . . . . . 42
31 Power Lost in a Sdden Difteaio, for a UniformVelocity Proms . . . . . . . . . . . . . . 47
32 A Family of Radial Velocity Distributions in aCircular Dut .. .. ........ . .48
xvW
Variation of the SbAp IntegraIs X and AP With theShape Exonent w for the Power Law Family of VelocityDistribations . . . . ........... 51
-"M Power Lost in a Sudden Dliusion, as a Function of theVelocity Distribution Profile . . . . . . . . . . 52
35 Variation of Velocity Ratio AIMSIA.,, With the PressureA.•se Parameter Cp , For Similar Profiles . . . . . 5&
36 Varlatloa of Total Pressure Loss With Static Pressure6Rise, For the Case of Uniform Velocity Dixtribution. . 55
37 The Elemental Geometry of the Curved Flow Field . . . 58
38 A Flow Which Has Infinite Radius of Curvature at theCenter Line . . . . . . . . . . . . . . . 62
39 Potential Flow Solutions by Gebbay Applied to the PlenumChamber Problem . . . . . . . . . ... . . 64
40 "Thin Jet" Geometry in the Abseme of Vinooua Mixing . . 65
41 Geometry of a Thiok Ainlar Jet With No VsoaM Mixn . 68
4a aen"of a c/z--_bwam .. . .. .To
43 rOmetrya~ lwmdT rGUa . . . . . . . . . 7048 Crosee-lt at Upelesm (1U0 M (180 To DetezmleVahee for the a*t OarvAM 1sxsewr (Mutton
brmme• -mwum• s ... . . .* 7744 Vaulsom of ad~v Fammiar It WUt tdo
45 Jet Thiowese at Anmtle Pramm , as a AFUMAJof tt4eNowgl Parameter 4*A (lamed an Coutmbo of masnFplow) . . . . . . . . . . . . . . 79
46 momentum Salome as a Fution of the HftefParameter / . . . . . . . . . . . . . . S6
3xi
47 Variation of Cushion Pressure With the HeightParameter . . . . . . . . . . . . . . 82
48 Discharge Coefficient CO as a Funt.tlon of the HeightParameter *. . . . . . . . . . . . . 83
49 Comparison of Theoretical Discharge Coefficiebts for0 =0 ................. . . . . . . . . 84
50 Comparison of Theoretical Discharge Coefficients forS = 30 . . . . . . . . . . . . . . . . 85
51 Comparison of Theoretical Discharge Coefficients for0 = 600 ........ .............. . . 86
0:2 Tot-l Nozzle roroe, Measured Parallel to the NozzleAxis, as a Function of the Height Parameter %4. . . . 88
53 Comparison of Discharge Coefficient Ratio •4 forTwo-Dimensional and Circular Planform GEMs (0 = 00) . 90
54 Effect of Planform on the Cushm PressureParameter I4A . . . . . . . . . . . . 91
55 The Anualar Jet Teot Rig With Ss Pawel Removed To
No No, C, ftivrim ft-ja•. 1.o 0 -300) . . . 101
56 Internal Dimensios of Annular Jet Tont Rig . . . . . 102
57 Velocity and Presm Distribution In the Throat of theAmmilarJet . . . . . .*. . . . . . . . . 104
58 Premsures and Velocity at Nozzle Exit 1. 00=300 XL=-.10nches * * * . .. . . 105
59 ledhstributlon of Power in Annlals Jet . . . . . . . 111
60 Static Pressure Variation Across a Jet - Comparisonof Theoretical and Experimental Values. . . . . . . 112
61 Pressures and Velocity in Annular Jet and Cushion.(Flat Cushion Board 1.0 0 . 300) . . . . . . 113
xX
62 Pressures and Velocity In the Annular Jet and Cushion.(High Cushion Board 1.0 = 300) ..... 114
63 Jet Total Head Distribution Needed to Give Constant JetVelocity . . . . . . . . ... . . ............. 120
64 The Cushion Pressure Parameter Aft as a Functionof the Height Parameter R/4t . . . . . . . . . 121
65 Comparative Effect of Constant Jet Velocity and Constant
Total Head on Cushion Pressure Per Unit Air Power. . . 123
66 Symmetric and Asymmetric Jet Total Head Profiles . . . 125
67 Geometry for the Jet Thickness Anomaly .. ...... 126
68 Two Rectangular Total Head Distributions . . . ... 127
69 Solution Limits for the Rectangular Total PressureDistribution ,-
70 Effect of a Stopped Rectangular Jet Total Head
Distrbution Upon Cushion Pressmre. . . . . . . . 130
71 Gmeral Linear Variation . . . . . . . . . 131
72 Efet of a Linearly Varying Jet Total Head DidtrbutionUponOushionPel seure. . . . . . . . . . . 132
73 Gemeral Power Law D-trto. . . . . . . . . 133
74 Cushion Pressure as Affected by a Liner Distribution ofTotal Head Ptesure in the Jet (Epomeantal Flow). . . . 136
75 Cushion Pressure as Affeoted by a Linear Distribution ofTotal Head Pressure in the Jet (Etonential Theory) . . . 138
76 Cushion Pressure Parameter With Linear Gradient ofTotal Pressure Across Jet - Free-Vortex and ExponentialFlow .* . . . . . . . . . . . . . . . . 143
77 Pressure Distributions in Jets . . . . . . . . . 146
xxi
78 Some Experimental Measurements on the Cushion 149Pressure Parameter at 0o .. .... . . .
79 Assumed Equivalent Linear Total Head DistributionAcross the Kuhn and Carter Model Jet .... ..... 150
80 Increment of the Cushion Pressure Ratio Due to TotalHead Distortion in the Kuhn and Carter Model. . .*. . 151
81 Data of Figure 78 With the Kuhn and Carter ResultsCorrected to Two-Dimensional Flow With UniformTotal Head .. . .. * . * * . . . * . 152
82 Curved Jet Flow Field in Ground Proximity. . .*. . . 15-'
83 The "Air Friction" Theory of Viscous Losses in aThick Jet Compared With Inviscid Flow Theory for
=00 . . . . . . . . . . . . .... 157
84 Effect of Nozzle Extensions on the Cushion PressureRatio(0 o00) . . . . . . . . . . . . . . 159
85 Nffoat of 2 aio/ng the omor krface of the Jet When
Nossle ZSamIca sArcUUsed . . . ... . . . . . 160
86 lefoad A-ze o- Data of Flogre 85. . . . . . . 161
S7 Mzi ias the AnslarJet . . . . . . . . . . .162
8 Two Way dof PozM• dto Overfd Jet. . . . . . . 1
so Theoroedia Loss Is Cushion Pressure Caused by
Viaoom sMz ( 0 a 0) . . . . . . . . . . . 172
90 of~ Vicm173TeryWt xprm.0 a a. 0 a 0 * 0 a 6 a 17
91 Prim• y Vortex Pressure as a FIunction of Hover Height(-o')0 so*$. .s. 175
92 Baic Geometry of Nozzle Dimsilon . . . . . . . . 176
xxii
93 Static Pressure Distribution on the Inside Nozzle Wall . . 177
94 Static Pressure Distribution on the Outside Nozzle Wall . • 1781
95 Approximate Total Pressure Loss Due to D/lfsion for*k = 1.0, in a Straight Nozzle. (Based on the ExponentialTheory Static Pressure Distribution). . . . . . . . 181
96 Mean Total Pressure Loss as a Function of the HeightParameter RI/ for a Straight Nozzde. (Based an Approxi-mate Exponential Theory Static Pressure Distribution) . . 183
97 Basic Geometry ............ . 185
98 Separation of Coanda Flow. . . . . . . . . . .187
99 Variation of WAll Pressure With the Jet TholwmassParameter . . . . . . . . . . . . . 188
100 Variation of Local Difst IMsiMol y With Lowal
VelocityRatio . . . . . . . . . . . . 0 . 19.
101 Flow spartonia Curv ket. . * , , . . 9. 6
102 Camels ble Uoft Owe Gomwd Thes DwehpdI.This , . . . . . .. . . . 186
103 Au dmed . . , , , , . • . .* .192
104 vaui"M .1 inw sad d am TOWNu WokdtAmpe ~te . . , . , . . . • • . .188
10o Poe#sM @d *S ofemw VuTsoy IWam e . . . . . Sol
106 naow of mass Fvw Ian l O b6 unu Comt V.oottyaSrenmla etothsTeOW W"eetlw . . . . . . .. 20e
107 Average Total Head Lass AremW a Oeeast DatThlickasSend. . . . . o. . . . . . . .206
108 Comparison Between Theory ad ZVpeurmet 1ow aRigft-Angl. Bend . . . . . . . . . . . . . S20
xxiie
zx~f
109 Flow Into a Flush Intake ........ . .208
110 Flow Into a Flush Intake . . . . . . . . . . . 209
III Variation of I With ZIw, for Uniform Inflow( A% = Constant). . ....................- 215
112 Variation of Inlet Velocity With Intake Lip Radius.(Exponential Theory for AAI )...... ...... 218
113 Variation of Inlet Velocity With Intake Lip Radius.
(Free-Vortex Theory fors, - 1. 0) . . . . . .*. 220
114 Arrangement of Intake in Floor of Wind Tunnel. . . .. 226
115 Location of Measuring Stations . . . . . . .*. . 227
116 Total anu Static Pressures in Intake. . . . . . . . 230
117 Velocity Vector Profile at Mouth of Intake . . . . . . 231
118 Velocities at Mouth and at Throat of Intake. . . . . . 232
119 Theoretdcal Velocity Distributions for the Test Intake(Unom Inflow) . . . . . . . . . . . . . . 233
120 Cross-Plat of Intake Velocity Ratio, as a Function of
4,IA'. hw the Test bUM. (jConservation of MomentumConeaatim Betweenm 7 adyl ). . . . . . . . 234
121 Cr*es-Pih Of bMa VelOt atio, as a Fwaotion of;60, ter tdo ?es* bake. (Comaerivation, of Mamie Flow
Coridelaim Dstlowssa andtj/.4,) . . . . . . . . 235
122 a of 2po•smeotal and Theoretical Velocityat lMoi . . . . . . .0 . . . . . . . . . 236
123 Reenmblae Between (a) Velocity Profile in UpstreamBoundary Layer and (b) Velocity Difference (Theoretical-Zxperimental) Near Upstream Wall of IntakeI Throat . . 2 236
124 Total Pressure in Upstream Boundary Layer ..... 238
xxiv
125 Spanwise Distribution of Velocity at Mouth of Intake . . . 238
126 Velocity at Throat of Intake - Tunnel Speed Zero and90 Feet/Second. . . . .... ............... 239
127 Jet Drag Due to Entrainment in a 90-Degree Jet Flap. . . 242
128 Some Effects of Nozzle Shape on the Discharge CoefficientUnder Static Conditions ..... ............ 243
129 Jet Issuing From a Body in Inviscid Flow . . . . . . 244
130 Boundary Layer Entrainment in a Jet . ........ 247
131 Jet Influence on a Wake. . . . . ... ........... 249
132 Nozzle Diffusion Losses . . . . . . . . . . .250
133 Jet Flap in Inviscid Flow .. . . . . .250
134 Smoke Tunnel Flow Observations of a Jet Flap ... 251
135 Local Jet Geometry. . . . . . .*. ... * * 252
136 Pressure Intepral on the Uj Btree 8id a jetxhausted Nornml tosFre-rim . . . . . . 256
137 Arragemedt of Jet D llms Through Floor ofWindTunel. . • . * • . • • .. . . 57
138 velocity Oad Pressur DwInW Dow-rem Flow JOt.(Jet/Free-tureeM Veleafty USo 0.W) . . . . .*. . 256
139 (a) Velocity sad Pressure l 1 t Mouth of Jet.(Jet/Free-Stream Velocity A ot S. 0W). (b) Velocityand Pressure Dlstr/iostia In Jet Throat. (Jet/Free-Stream Velocity UAtjo 0. 4) . . . . . . .. 269
140 Velocity and Pressure Distribution Downstream FromJet. (Jet/Free-Stream Velocity Ratio 1. 34). . . . . . 260
xxv
141 (a) Velocity and Pressure Distribution at Mouth of Jet.(Jet/Free-Stream Velocity Ratio 1.34). (b) Velocity andPressure Distribution in Jet Duct Throat (Jet/Free-StreamVelocity Ratio 1.34). .. . .. . . .. 261
142 Velocity and Preshure Distribution Downstream From Jet.(Jet/Free-Stream Velocity Ratio 2.88) .... ...... 262
143 (a) Velocity and Pressure Distribution at Jet Mouth. (Jet/Free-Stream Velocity Ratio 2.88) . . . . . . . . 263
(b) Velocity and Pressure Distribution at Jet Throat.(Jet/Free-Stream Velocity Ratio 2. 88ý .. ........ . 264
144 Apparent Effect of Reynolds Number on the CushionPressure Ratio . . . . . . . . . . . . . . 275
145 Effect of Nozzle ftn Friction Loss on Apparent CushionPressure Ratio . . . . . . . . . . . . . . 276
146 Geometry of a Two-Dimensional Vortex. . . . . . . 278
147 Total Pressure Los at the bidt Pk me of an AumilarJetNonle , . ............. 282
X01
SYMBOLS
The prefix A denotes an increment of the appropriate quantity. When itprefixes a pressure, it denotes that the pressure is "gauge"; that is, thepressure is measured relative to ambient static pressure Pa.
E.g., &P P- P
In general we employ the conventional X and V axes wherever possible.In different parts of the report the same symbol is sometimes used to denotedifferent quantities. This is unavoidable because of the large number of dif-ferent topics discussed, but the precise meaning is always made clear in thetext when this duplication occurs.
Where special symbols are defined and used in only one place in the report,they do not appear in the following list.
A an area
A¢ cushion area
e' periphery or lmgth of a jet
, coefficient of discharge
or a a drag coeffliient
skin friotion ooeffioit ao a solid bwundary
Sapparent skin friction coseficient at a fluid interface
(. a pressure coefficient ( )d4~4.Ir coefficient of local skin friction stress
.ID diameter of a circular planform onM
-"11
F a force
Fw total nozzle thrust
.• height of an annular jet nozzle above the ground plane
or 4 a dimension specified in Figure 23
a total pressure loss
T a momentum flux
(le.w &I , for example)
-TI4 nozzle momentum flux
K a constant
Of an exponent
A a mass flow rate
At etrained air ntio
P a statw proem"o
P a towalpresmure (~r4 # for example)
V a pow=r
• a cushion presmure parster defined in Equation (142)
a dnaic presur ( fmor example)
e a radius, usually the local radius of a curved Jet
eo inner radius of a jet
Router radus oc ajet
Xxvw
t nozzle width or jet thickness
I thickness of a jet at ambient static pressure
44 velocity parallel to the X axis
W. free-stream velocity
A.v a mean velocity
A)` velocity parallel to the I axis
or 4P a resultant velocity
W width of a rectangular duct
2 a height parameter z f& . s')
or W distance along the Jr axis
a height paxamoter a I a+ - f MD)
distance aluag the ýfaxis
distance across a Jet
& an elemental mass
Sa flow curvature param eter
or r an efficiency
I angle at which a jet is inclined Inward from the vertical
a velocity profile shape factor defined in Equation (48)
1A4 coefficient of fluid viscosity
XXix
mass density of the fluid
"1 local skin friction stress
San augmentation ratio (total thrust/primary thrust)
Sb a velocity profile shape factor defined in Equation (53)
or # a parameter defined in Equation (258)
xgx
I
Chapter One
SUMMARY AND DISCUSSION
Although It is more usual to have a "Discussion of Results" near the end of aresearch report, the diffuse nature of the present progra-. seems to render adifferent approach more appropriate. Accordingly we shall review the programin a fairly general way in this chapter, in order to place it in a proper perspec-tive. Subsequent chapters will deal with the details of the various investigations.
THE FUNDAMENTAL PROBLEM
"Underlying the whole of this program is the central phenomenon of a subsonicfluid flow field which is not rectilinear; that is to say, one which is constrainedin some way to flow along a curved path. Some examples of practical problemswhich involve such flows are sketched in Figure 1,and it is evident that our studyis not lacking in practical utility.
Until quite recently, practical aerodynamic theory was almost entirely basedon the assumption that air was inviscid. Viscous effects were considered onlyto arise in the boundary layer, as first suggested by Prandtl, and simple meth-ods were evolved for applying viscous effect "corrections" to the various in-viscid flow theories. The reason for this was that the viscous shear stress ina fluid is proportional to the local velocity gradient; that is,
stress M
so that, if •is• in small, the viscous fore" are negligible. Prandtlshowed that the velocity gradient M negligibly small in practical flow fields,except for a very thin layer of fluid close to a solid boundary. Thus, by re-garding this boundary layer as part of the body, wo can use inviscid flow theoryto determine the main flow field. TIh details of the boundary flow can then betreated as a separate problem; most usually, in fact, we do not need to studythe boundary layer, but rather should represent Its effect by an empirically de-termined "skin-friction" loss coefficient.
The foregoing might appear to relegate the boundary layer to a position ofnegligible importance in engineering serodynamics. This is by no means so,of course, since it is responsible for most of the drag of streamline bodies andalso is the controlling factor in separation of the flow ("stall") from the surfaceof a body. Until quite recently, however, and excluding some special cases,these problems have 'een dealt with on an entirely empirical basis.
1
(a) Flow into a horizontal (b) Annular jet flow.intake.
(a) Flow in a c-.rved duct.
(d) Coanda let flow.
Figure 1 Curved Flow Fields Considered in This Report.
2
A jet is a stream of air which has a velocity markedly different from the mainbody of air being considered in a problem. A conventional jet is moving fasterthan the surrounding fluid, but it is no different, in principle, from the "negativejet flow" into an intake, Figure 1(a), or the wake behind a body. In all cases thereexists a substantial velocity discontinuity at the surface of the jet so that, fromEquation (1), viscous forces become important.
Jets did not play a significant role in aerodynamics until the advent of the gasturbine for propulsion; even then, it was the propulsive force obtained, ratherthan the finer details of the jet flow structure, that received most attentim. Thepropulsive force depended on the flow condition at the engine exit nozzle, not onwhat happened downstream.
The advent of the jet flap, annular jet, and other similar concepts markedlychanged this picture. For the first time, a jet was being used to influence themain flow field, and the characteristics of the jet after it left the nozzle werefound to be the controlling parameters.
Early investigators tried to tackle these new problems with the well tried in-viscid flow theory techniques, not only with limited success, but sometimeswith nonsensical results. The famous "thrust hypothesis" for the jet flap isa good example of this.
- Wo . . --
tD
Figure 2. The "Thrust Hypothesis"t for a JetFlap in Inviscid Flow.
3
Davidson's original hypothesisI is illustrated in Figure 2. If the control bo, nd-aries are drawn far enough away from a jet-flapped wing,it can be shown thatthe horizontal momentum flux out of the box is greater than the flow in, by theamount 3*. Thus, although TT is a vertical vector at the wing, we must ex-pect a horizontal (pronulsive) force ZJ. to appear on the wing.
In inviscid flow, both Davidson and Stratford2 assumed -To = T , so thatall the nozzle jet reaction should appear as a thrust force on the wing.
At the risk of being a little pedantic, perhaps, we should note that the totalnozzle force ( F" ) is greater than •. The correct relationship for in-viscid flow is
Fs4 #(2)
where = mesa static preseare at the nozzle
PM = jet total head at the nozule.
Equation (2) is always somewhat less tha u/ty, so that the "thrust hypothesis"is only approximately correct, even in inviscid flow.
In a real fluid, free-stream air mixes with the jet. Payne has shown that,when such mixing takes place at a pressure which is greater than ambient, thefinal momehtum flux is always roduced. In other words, the loss cf free-streaiair momentum is greater than that gained by the Jet. Calculations show that theredactiom in rM caused by this effect adequately explains the measured "thrulose" of a }et flap at high defleotion angles, even though there are some small aditsW losses (srck "s nozzle diffusion loss) which will be identified in subseqipostlms e9 this repo01.
We conclude, therefore, that jet flows cannot, in general, be explained by in-viscid flow theory. Thus, a large part of the present program has been devotedto developing methods which allow viscous effects to be calculated.
Curved Jet flows present the same general problems as rectilinear jets butwith the added complication of centrifugal acceleration effects, as indicatedin Figure 3.
4
FORCE Lr IL
Figure 3. Centrifugal Force in a Curved Jet Element.
Since the centrifugal forces must be balanced by a static pressure gradientacross the jet, even the Inviscid flow field is fairly complicated in this case.Although many attempts have been made, we know of no completely satisfac-tory solution to this problem, so that development of an inviscid curved Jetflow theory was the first task undertaken during this program. This wasthen used as the basis for corrections due to viscosity. Largely becauseof the terms of the initial work statement, this theory has been most &ullydeveloped and experimentally confirmed for the annular jet case. This hasno particular significance, of course, and the general theory, with appropri-ate end-conditions, is equally applicable to any of the problems sket•cbed inFigure 1. It isbelieved to be % powerfil tool for the solutien of a very largeclass of subsonic flow problemo, and it was naftrally impossible to developits full potential in the present program.
DIFFUSION LOSSES
A viscous fluid can be acoelerated without ihcurring Important energy losses,so long as compressibility effects a not enoountered. The process of slow-ing down a fluid -- diffusion -- always involves large energy losses, however.Thus,diffusion, and its associated phenomena of energy loss and flow separa-tion, constitutes perhaps the most important problem in subsonic fluid dy-namics. Paradoxically, it is a problem which has received relatively littleattention, so that today the position is the same as it was decades ago, atleast so far as minimizing diffusion losses are concerned.
5
FORCE = ' -•
ttf t,
Figure 3. Centrifugal Force in a Curved Jet Element.
Since the centrifugal forces must be balanced by a static pressure gradientacross the jet, even the Inviscid flow field is fairly complicated in this case.Although many attempts have been made, we know of no completely satisfac-tory solution to this problem, so that development of an inviscid curved jetflow theory was the first task undertaken during this program. This wasthen used as the basis for corrections due to viscosliy. Largely becauseof the terms of the initial work statement, this theory has been most fullydeveloped and experimentally confirmed for the annular jet case. This hasno particular significance, of course, and the general theory, with appropri-ate end-conditions, is eqaally applicable to any of the problems sketched inFigure 1. It sbelieved to be * powerftl tool for the solutio of a very largeclass of subsonic flow problemij, and it was naturally Impossible to developits full potential in the present program.
DIFFUSION LOUSES
A viscous fluid can be accelerated without iicurring Important energy losses,so long as compressibility effects are not encountered. The process of slow-ing down a fluid -- diffusion -- always Involves large energy losses, however.Thus,diffusion, and its associated phenomena of energy loss and flow separa-tion, constitutes perhaps the most important problem in subsonic fluid dy-namics. Paradoxically, it is a problem which has received relatively littleattention, so that today the position is the same as it was decades ago, atleast so far as minimizing diffusion losses are concerned.
5
Needless to say, the static pressure gradient generated in a curved Jet flowmeans that diffusion must occur at some point. Hencewe were very concernedto find some way of analytically predicting the associated losses.
Even before the present program began, we noted two peculiarities. The firstof these was the empirical observation (References 5, 6, and 7) that diffusion ispossible only down to a velocity of about half the initial value, unless somemeans of renewing the boundary layer is used (multistage diffusion, boun-dary layer suction, etc.) The second point of interest was the fact that sim-ple "Borda-Carnot" momentum theory for a rapid diffusion in a pipe gave verygood agreement with measured losses.
Figure 4 Borda-Carnot Diffusion.
The Borda-Carnot theory tended to be an isolated theoretical oddity, out ofthe main stream of fludd-dynamic theory, but we felt that it might offer im-portant clues to more eneral oases.
In developing this theory In a more general form,we found that it tied in withthe first (empirically observed) peculiarity In a very surprising way. If the"velocity profiles before &ad after diffusion were mathematically similar, thenthe theory showed Us the mean velocity ratio could not beless than 0. 5. Ifthe velocity profiles were dissimilar, then the maximum theoretically attain-able difsion varied In the same manner as Indicated by boundary layer theory.
Obviously, it would have been desirable to follow up this clue, since we maywell be on the threshold of a "unified theory" of diffusion. Unfortunatelythelevel of effort required was not possible under the present contract. However,it was possible to feel some confidence in the generalized (two-dimensional)diffusion theory produced during this part of the investigation, even though
6
its derivation, and certainly its application to some of the curved flow prob-lems, could not be described as rigorous.
An interesting example of the application of the diffusion theory is the curvedduct problem of Figure 1(e). The curved flow theory shows that, at the startof the bend, there is a sudden increase in static pressure on the outside walland a reduction on the inner wall. At the end of the bend the situation is re-versed, so that two diffusions occur: one on the outside wall at the beginningof the turn, and cae on the inside wall at the end of the turn. By using the in-viscid theory to calculate these pressure distributions, and then applying thenew diffusion theory, we can calculate the total head lost in the turn. Notsurprisingly, as shown in Chapter Nine, we find that, for two-dimensional flow,
Total head lost duct widthinitial dynamic head - radius of curvature
Although no two-dimensional flow test data is available for this case, we findthat extrapolation of the available three-dimensional data gives excellent agree-ment with the calculated theoretical loss. This obviously opens the way tosignificant Improvements in the design of low-loss duct beands.
We have also applied this diffusion theory to the calculation of the total headlost in an annular jet nozzle. O(we apain, the agreemet with experlment isgood, both so far as total loss is conoermed and for te loss distributionacross the duct.
When applied to d after a Coaus flow, the theory gives rather largerlosses than for an oqaivalest pipe bead, but mhsbatuiy, we kow of no ex-perimental data which cm be usd for a - Pa"Moses in this case.
In summary, the mw dimftdoa theo dow-plod -1 di the course of thisprogram is apparently capabl of dealing wib reltively sadden pressurerises in two-dimensionml flow sad can ( smbly) be eotemdd to three-dimensional flow problems. A very lited comparison with experlmentindicates good agreement, but abstantidally more work will have to be donebefore we can feel confident of It, beoause its basic derivation is not mathe-matically rigorous, at least for curvilinear ar laons.
In the following sections of this chapter we shall briefly summarize the basicwork done on viscous mixng, curvilinear flow, and diffusion loss in the prac-tical applications considered in this report.
7
ANNULAR JET THEORY
As for all of the problems considered, the basic differential equation forcurvilinear flow applies to the annular jet problem. This equation is
dev. 41 1(3)
Here is the local static pressureis the local total pressureis the local radius of curvature
In the following paragraphs we shall describe the solutions to this equationobtained by the investigators who pioneered annular jet theory. With thebenefit of hindsight, we shall derive their solutions, from Equation (3), inone or two lines of analysis. This in no way detracts from their achieve-ments, of course, because it is always easier to follow than to find a way.Also, it is doubtful that they realised that their problem was really thesolution of Equation (3); rather, each appeared to formulate the physicaldescription of the problem in a less fundamental way. Only in the presentinvestigation has it been realized that Equation (3) is the starting pointfor all the formulations.
We shall also Ignore, for the time being, the potential flow solution ofStrad 1 4 , sinoe this is based upon am entirely different analysis. Strand'swork was by far the most sophisticated attack on the annular Jet problem,of course, and best agres with the theory developed in this report overmost of the operating belht range. It suffers from a serious flaw, how-ever, in that it deviates from kow potential flow solutions as the nozzleapproaches the ground plane, a defect which the present theory avoids. Inthe limit 41V -P , for meaple, the discharge coefficient approaches0.5, instead of the known solution of approximately 0.62 for a slit.
The reason for this is not known. Strand himself, in verbal discussionswith the senior author of this report, could only suggest a possible reason;thus an attempt to explain this anomaly is hardly likely to be fruitful.
8
Mn (i1susIfing inviscid flow solutions to the annular jet problem we shall notcurpare results with experiment. Many investigators have shown that real(viseeous) annular jet flows generate relatively strong secondary flows, soOiatt there is ohviously a significant loss in cushion pressure, relative to theideua inviscid fluid case. Also, as shown in the present report, there is aslnificant nozzle diffusion loss. Thus any appeal to experiment, as a means1d judging an inviscid flow theory, is obviously likely to be unsatisfactory. In-
deed, wc might almost go so far as to say that an inviscid theory which agreedwith exxperiment must be wrong, since the theory should not contain the viscousmixing and diffusion losses.*
S!11APLIN'S SOLUTION (THIN-JET THEORY)
911ic first i ..... annular jet thcory %was Chaplin's , of course, where this
Equ-ation was expressed in finite, rather than infinitesimal, terms. By taking-as a constant ( = • say), and assuming that the static pressure varies
inur'.r~ across the jet, that is,
(4)
then Chaplin obtained, in effect,
,c common with later investigators, he also assumed that the jet total pressure.... was constant ( AP , say) across the Jet.
'tihs at the inner streamdinc (1 , ),
t- 2. 1ae (5)
iL, then calculated the radius of jet curvature ( R ),using the thin jet geometry:-howvin Figntre 5.
!Since writing this report we have seen an excellent paper by Eames 2 7 in which11 i oi't is emphasized.
9
ray
JmFigure 5 Chaplin's Curved Jet Geometry.
From this geometry, the clearance height (i) Is evidently given by
I.. R -0
W -6+S4) a (say). (6)
Thiu Cbap~als relaslaoblp for the oushion pressure (AhIj) becomes
-a Z O, (7)
It is found that this gives fair agreement with experiment for small values ofx (large hover hel~hts) but becomes progressively less accurate as theheight is reduced. When the vehicle is touching the ground ( ka o ), Equation(7) gives A,&/IP1 = 2. 0, whereas we know that the correct answer shouldbe 1. 0; that is, because no air Is flowing, the total and static pressures areequal.
10
Chaplin recognized these limitations, of course, and specifically limited histheory to "thin jets" in which -A > t .
As mentioned above, Chaplin assumed that the jet radius of curvature wasconstant. To be more specific, he assumed it was constant along *he jetbecause it divided two constant pressure areas: the "cushion" and the am-bient air outside. He was Wble to show that the assumption of constantpressure boundaries of necessity required a constant curvature.
He also assumed that the curvature was constant across the jet. This is(mathematically) acceptable for thin jets, but gives rise to a rather mean-ingless physical picture, since the jet vanishes at the ground plane. Thisis illustrated in Figure 6, a thick jet being shown for clarity.
R R
aJtVanishes
Figure 6 PysIal IotoerPtSIon of the ConstantRatus od Osralre Jet.
THE COSU-S-BTANTON-JONE5 SOLUTION I(XPONENTIAL THEORY)
Crosn10 and Stanton-Jrensu1 (the prcedmce cannot be determined fromthe literature) extended Chaplin's theory to the oe of "thick jets". Theironly change was to reject the assumption of constant static pressure gradi-ent across the jet, Eqiation (4). Blnoe they retained Chaplin's constantradius assumption, their version of Eqaation (8) was
IL
Or, since *
-A (9)
This is a simple first-order differential equation with constant coefficients,and the solution is well known. That is,
When we insert the end condition AP; = 0 at 0' = 0 (jet static pressure the
same as ambient at the outermost streamline) this equation simplifies to
J11Ali I~e I(111
and the cushion psrsuure, which corresponds to the innermost streamline(y, i"). is then
AR~ (12)
This equation tends to unity as the ground clearance height (4 ) tends to zero,thus avoiding the most mportSmt limitation of Chaplin's solution. When com-pared with ereeriseual u it gives quite good agreement. As a result,the general theory wich follows from 2quation (11) is widely used in the in-dustry. Yet if this agreement Is fortuitou, acceptance of exponential theorycould prevent the obtaining of a better physical picture of annular Jet flow,and heoe prevent the discovery of more efficient ways of generating an annularJet.
PINNZS' SOLUTION (rREE-VORTX FLOW)
Exponential theory ts based upon an impossible physical picture, as we haveseen, since the Jet cannot possibly vanish as it strikes the ground. Pinnes12
chose to circumvent this by assuming that the Jet thickness remained constant,as shown in Figure 7.
12
A,
7 f 7-7- 7 7 /
Figure 7 Pinnes' "Free Vortex" Assumptionfor Annular Jet Flow.
We can immediately see that this formulation must be to a "thin-jet" theory,since if X < t , the outer radius ? will have to be negative, a physicalimpossibility.
While retaining the assumption that any jet streamline will have constantcurvature along its length, the Pinnes formulation assumes that (f )will vary across the Jet in a linear manner. That is,
-o ' M +; (1 3 )
This is a significant improvemed in rigor relative to exponential theory.The basic differentil equatino now becomes
2 " 2)tI (14)
The standard form solution then gives
! a /3 (15)
13
and the cushion pressure ,.
- Z (16)
In this case /- = 1. 0 when • 1. 0, a result which is obviously in error:due to the geometrical limitations noted above. Also, as X--* 1.0, the theo-retical mass flow tends to zero. Thus,we have to conclude, rather paradoxi-cally, that although the Pinnes formulation is more rigorous, it results in aless accurate description of annular jet flow. In the sense that it pointed theway to further developments, however, the Pinnes theory represented an im-portant advance.
PAYNE'S CORRECTION FOR JET CURVATURE
All the previous theories used Chaplin's relationship (6) for jet curvature.Payne 13 pointed out that this was correct only for zero jet deflection angle,or when the jet was of negligible thickness. The correct relationship for ajet of finite thickness (t ) is actually
e'- (17)
so that (say)~~ 18
in place of Chaplin's relationship given in Equation (6). In Reference .',Payne showed that the available experimental data correlated better againstthis new parameter X
THE ANNULAR JET SOWTIOM OF TEiS SEpORT
Although Pinnes improved the pysicWal representation of the annular jet atlarge heights, be stopped shwrt at realizing a perfectly general description.As indicated in Figure 8, his gemetry is approximately correct at largeground clearances and the eapsatal theory (constant radius) geometry isapproximately correct for very low cleauanes.
14
(a) Low Clearance.
(b) High Clearance.
Figure 8. Effect of Ground Clearance on JetRadius of Curvature.
At low heights the static pressure in the jet is high and the mass flow is low.rhus, after the jet has expanded to ambniet pressure along the ground plame,it is moving much hfstr, and ito thiohens ( a ) is mmoh less than the nozzlethickness (it). Under thes omndii•lm, 40 Is omly a little less than R, andexponential theory gives a good dsowrptica of the flow.
At high heights, there is little difforeme behtem the cushion pressure andambient, so that It., t. Under these oeeotions there is little change injet thickness and Pinnes wote gives a pod description of the flow. Thusthe two limits are
je as 4~.I
We naturally wondered if it would be possible to Interpolato these two limitcases with a more general variation.
15
When I - 0, we have exposential theoiy mad w 1m 1. 0 we havefree-vortex theory. In general, v will be between these two extremevalues.
As will be shown later, it turns out that such an itterpolation is possible,and we [ind
.50i. 49(21)
Thus the local radius in $be lot is mo given by
1.~t / '476 (22)
Epaatioa (3) the b inmama
e+ ?;) (A# ;) j 23)
and the ginral sehlon in IN A-.
4%~(~."7) ~(24)
Note that In this eaation we have retained Atlet an arbitrary function of(2 ), rather than assume it to be oonstant, ?s previous workers have done.Ve shall find that a varlation in~ across the Jet Influences the cushionpressure generated and that this din ba quite different from the result ob-tained by assuming a constant total pressure equal to the mean value.
16
1.0
PRESENT THEORY= 600
,08 .0
FREE-VORTEX FLOW.0
0.6 EXPNENTIAL THEORY
Z 0.4
0.2
0 0 2 4 6 8 10
HEIGHT PARAMETER
Figure 9. Comparison of the New Inviscid FlowTheory With Previous Theories.
17
1.0
S
0.8
S0.6
0.4
0.2
0 t _ _ _ _ _ I _ _ _ _ _ I I___ _
0 2 4 6 8 10
HEIGHT RATIO -l
Figure 10. Compariscn of the New Inviscid Flow TheoryWith Some Experimental Data (0 = 0°).
13
For the case of uniform jet total pressure, the new theory is compared withexponential and free-vorte,. theories in Figure 9. As we should expect, itfalls between these two limits, tending to the Pinnes solution above R.* = 4. 0.In the intermediate range,it gives a cushion pressure which is quite significantlygreater than that given by exponential theory.
Referring now to Figure 10, which shows some experimental data in compari-son with the new theory, we may wonder whether the new theory was worth de-veloping, since all the data points fall below it! The theory is for inviscidflow, however, and we shall see that when we correct it to allow for the effectsof nozzle diffusion loss and viscous entrainment in the jet, good agreement withexperiment is obtained. The older theories disregarded these effects and henceoffered no rational methods of minimizing them.
MOMENTUM BALANCE OF THE ANNULAR JET
An important offshoot of the analysis presented in this report is the concept ofmomentum balance.
•F f r OP .7 7 7" ff F f P
Figure 11. Momentum Balam. Geometry.
Referring to Figure 11, it will be shown quite rigorously that, if R Is aconstant, then
where J,- is the momentum fluxto ambient and 44 is the effective Jet periphery(jet length in the two-dimensioual case).
19
This simple relationship is of considerable value, since It Is quite fundamental.X
COANDA. JET THEORY
(a) Annular let flow (b) Coanda jet flow
Figure 12. Equivalence of Annular Jet and CoandaJet Flows.
As indicated in Figure 11, there is no difference, in principle, between theannular Jet, which is curved outward by the cushion overpressure, and theCoanda jet flow, 'which is curved inward by wall "suction". Equation (3) isstill applicable but the "end condition" of ambient static pressure is now trans-ferred to the opposite side of the jet.
In the present investfpatiom we have considered only the free-vortex solution(4b .SO ÷P ) p oUh as More gparalceof( Cs =f 4 47 may be moreoapplicable when the jet thlolmess is large.
One basic differsee in the overall flow system is the diffusion which occursat the ed of thec urved flow path (IK-X in Figure 12b). If the curve intersectswith a tangential stralgM wall, then the sudden disappearance of the centrif-ugal forces in the flow must result in a sudden static pressure rise at the in-tersection. This effect is amenable to the diffusion loss analysis mentionedearlier, and we find that quite large pressure losses are to be expected whenthe Jet is thick. Coupled with the skin friction loss and the static pressuregradient across the Jet, this diffusion loss explains why laboratory studies ofthe Coanda effect do not show any thrust augmentation effect, even though thejet entrains ambient air at a lower-than-ambient static pressure.
20
In Reference 3 (part of which was written under the present contract),ft wasshown that thrust augmentation occurs when a jet minss with ambient air at astatic pressure which is lower than ambient. As indicated In Figure 13, thereis an optimum static pressure for maximum augmentation; mixing at pressuresaway from this optimum results in less than optinmum augmentation.
AUGMENTATION
-AppA
Figure 13. Variation, of waoi o • Withthe Mixing Prom" Parameter 4.
Thrust augmentation is deffsed as
-Towsltbzsccf00NooTnsti pof KNOW e etosif to Aui-iet
The mixing pressure parameter is
i- ttc presaien 9Which stm- occrsI"" ru~mc hed ofprimary jet
In Coanda flowthe mixing pressure varies from ambient ( a =0, point Ain Figure 13) at the outside of the jot to
-17 W- - (26)
21
at the wall (point B in Figure 13). Thus,the mixing pressure can only be
optimum at one streamline (•.),and the overall augmentation will be less
than optimum.
A second factor determining augmentation is the efficiency at which the flow
is diffused back to ambient from the mixing pressure Ap, . A typical variation
is sketched in Figure 14.
X's
Diftsion efficiency
Figure 14. Variation of Augmentation Ratio WithDftsiso Efficiency, Assuming OptimumMixing Pressure.
It is obvious that woitwhlle thrust Increses can be obtained only when the
diffser efficlawy Is hi - 30% or greter, say. For relatively thick jets,
the ditasice les at the end of a Coanda curve can be quite large -- % as
low as 0. 5 -- so that high mudmemtation ratios can be obtained only by keeping
the jet thin. The loass du. to skin friction then become important, however.
Another reason for using thin Jets when augmentation is required is the amount
of ambient fluid entrained in the jet. The attainable augmentation ratio naturally
increases with the amount of fluid entrained and, as indicated in Figure 15, this
becomes important only below 4/4 = 0.1. Very high entrainment ratios re-
quire a jet thickness which is only a few percent of the radius of curvature;
and in this case, boundary layer effects start to become important.
22
COANDAREGION
1. 2
1.0
0.8 -
I 0.6
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5
JET THICKNES PARAMETER tlP,
Figure 15.. Variation of Entrainment Ratio With Jet
Thickness for a Right-Angle Coanda Bend.
23
Although nothing can be done about Coanda mixing at nonoptimum preamree,we can avoid the efficiency loss due to diffusion, as indicated in Figure 16.
I *. COANDA FLOWI ) * SECTION
I / ,
( I DIFFUSIONSECTION
Figure 16. A Coanda Profile Designed for MaximumThrust Augmentation.
In the present program vw have teaded to regard the Coanda effect a an in-teresting test of Ow thetortoal mehods as they were developed, so that theamount of ooheremtly commoted sldy of this phenomeno was rather slight.The basi teohaiqipe, however, would seem to satisfy all the requirementsfor an ezhiostives aft. We believe that suoh a study would be very fruitfuland might lead the way to Importat fure applcations of this interestingpenoomenon.
WNrAKE FWW TFLOW
The intake flow field depioted In Figure 1(a) Is beooming increasingly impor-tant in hardware applications, some euimples being the XV-4A, XV-SAo andvarious GEM onfiguratio.
Since it is a curvilinear flow problem, Equation (3) again applies; and in asubsequent chapter of this report, the appropriate end-conditions are defined,and solutions obtPined for certain simple cases. Characteristically, the air
24
I
accelerates over the forward lip and then diffuses into the duct, so thatseparation of the flow can occur, as indicated in Figure 17.
Separation due to suddenpressure rise.
Figure 17. Leading Edg Separation With Intake Flow.
Even without separation, total pressure losses occur, and once again, thesecan be calculated with the diffusion theory developed dring this progrm, al-though lack of time bas prevented us from actually doing this. Tlus, the thoo-retical conoepts developed present us with the meas of designin optinumintake geometries.
The classical approach to each problems is to use the method of singularities tocalculate the invocsd flow field. This bas drawbado. of course, because a realfluid tends to separate in 1ia$ c' high adveree pressure gradient. Moreover,the fact that a real flow Is aot at a conitt melW level proeludes such an ap-proach from ever giving preoli, results unles the oofisuration under oonsid-eration is one In which constant meol is a viablo assunqti.
In a practical intake flow problem, It Is usual for some boundary to exist up-stream of the intake opening. It will be shown that even a very thin boundarylayer can have a very marked oefht on the intake flow distribution. Moreimportantly, the separation or flow distortion which results from high dragareas upstream of the intake can have a large influence upon the intake flowdistribution and on whether diffusion separation of the flow will occur.
25
These effects can be studied, using the techniques developed, but once again,
lack of time has precluded such studies in the present program.
MOMENTUM DRAG
The momentum drag of an intake is usually defined as
Drag - (mass flow) x (free-stream velocity)
Ad*V . (27)
Various investigators have reported anomalies, in that the difference betweenthe measured profile drag (air off) of a vehicle -- particularly a GEM -- and thetotal drag is less than (A U, ); sometimes there is no difference at all.
The present work on intake flow indicates that Equation (27) is not correct,since the intake is not swallowing undisturbed air in most cases. If there is asubstantial skin surface upstream of the intake, for example, the boundarylayer air has already been slowed down to a velocity less than O/o . If thereis a high-drag element (such as a stalled wing leading edge, or a bluff bodyshape, upstream) then all the intake air may have an initial velocity which issubstantially less than ,4 . It Is felt that this probably explains all theobserved anoamlies, altho,.gh it has naturally proved impossible to under-take a detailed review of them.
T JT 71AP
The 'thrust hypodh•eis" of the jet flap was earlier referred to and illustratedin FIpgre 2. In aa effort to obtai some firo-hand experience, we built a 20%thWok llipIal &er u ot the Pay.. two-dimensional tunnel, and also ran asmtilar model in the smoks tunel.
Typioal msi. tUnmel runs are llustrated in Figures 18 and 19. These arequite oooveutloal reults,'exoept for the indication of turbulence in the wakeof the wing. It is also very Interesting to note that, below the wing, the free-stream air eotors the jet, rather than being deflected by it, as would be expectedin invisold flow.
We then studied the forces generated in the two-dimensional tunnel; using thetest setup Indicated In Figure 20. As indicated in Figure 21, operation of theJet flap generated more drag, rather than a thrust force !
28
IL i'1
27
0'4.40�.4
-4.
iiSI.e
.1ra4
28
29
* SEALED NOZZLE4 •j/4. =1.7
-0.3
-3 0.2
o /04
0.1
05 6 7 8 9 10
REFERENCE POSITION ACROSS WAKE - INCHES
Figure 21. Results of Total Head Traverses 3.25 ChordsBehind the Jet Flapped Aerofoil, With the JetFlap On and Off.
30
The main reason for this is indicated in Figure 22. Because of the relativelylow jet to free-stream velocity ratio, the jet entrainment is insufficient toavoid flow separation around the trailing edge of the wing. Thus, the jet sheetactually *.acreases the dopth of the wake.
Given a relatively slow-speed wake, the jet now loses more energy by mixingwith Jt, and also by mixing with free-stream air in the region marked X-)( inFigure 22.
The net result was that, instead of obtaining
:7o
as Figure 2 would indicate, we measured
40 = - 0- o3
;70
Presumably Z./3,- would have been positive if the Jet had been energeticenough to prevent flow separation at the trailing edge. Mixing in the regionx-x would still have occurred, however, and In Reference 3 it Is shownthat even this is enough to explain the low values of thrust recovery( =V% - *.1- 04 ) reported by other workers.
JET IS•UING NORMAL TO A FRZZ-UTRZAM FLOW
The Jet-flap measurem t Indicated that the ftsioal picture was extremelyccmplicated. In an effort to Isdt some of the variables we therefore do-signed an experiment in which a Jet issued from the tunnel floor, as illustratedin Figure 23.
From Equation (25) we should expect the balance of momentum flux and thepressure forces to give
(28)
31
TRAILING EDGE PROFILE
-I -4
WAWAKE~~v;~
3203
if Aý!. -
and since
(29)
5,. * * .<' ,:: . :.'• •;
the following
33 ti t
Fiur 23. . Asne Geme.iyTh paraete (/ \.r--- -;.)i tde ntemanbcyo hsrpr
and• Isshow to : b-e... --- ..... , isni etojt hp.-.-For th test codtin employe in the. "-,----ina u e baieth .~o i n" . • ,.. -
! -," " ,I .- /! I• 11'33/
--/410 .56 1.34 2.88Measured --- 5.0 7.5Theoretical 0.94 75.0 169.0
The theoretical figures are based on the assumption of a constant radius curve,but even in the limit case they would be reduced by only two-thirds. Obviously,the theoretically predicted -/1& is an order of magnitude greater than themeasured value.
A small part of this is attributable to assuming • = 0. From the test datawe can see that * < a , due to viscous entrainment of the cavity air. Mostof it must be due to a reduction in • , however; part of which is attributableto mixing with the cavity air and part to mixing with the free-stream air.
Although some aspects of this problem have been studied briefly in Reference3, we have not attempted to formalize the theory completely for this problembecause of lack of time. Nevertheless, sufficient work has been done to indi-cate that it is probably amenable to the general techniques developed duringthis program.
34
Chapter Two
GENERAL DIFFUSION THEORY
THE FLOW OF A JET THROUGH A DISCONTINUITY IN STATIC PRESSURE
The flow of a jet through a change in static pressure is not well understood,perhaps because it rarely occurs in practice; or rather, it was rare untilthe advent of the jet flap and the annular jet. A one-dimensional solutiondue to Borda and Carnot is known for the case of an abrupt duct enlargement(the "Borda-Carnot loss" or the "Carnot impact formula") and gives very goodagreement with experimental observations. However, this analysis has notbeen extended, and it is the purpose of the following sections to generalize thetheory and to extend it to two-dimensional flow.
I T
4,--am.i
Figure 24. Geometry of Flow Through a Pressure Change.
SOLUTION FOR CONSERVATION OF TOTAL H19AD
In Reference 4, Payne assumed that, in the absence of a mechanism to accountfor losses, total head is conserved when a jet undergoes a change in staticpressure. Referring to Figure 24, this amounts to assuming Bernoulli'secpation for conservation of total head to apply along any streamline
3(30)
35
For the one-dimensional case, the momentum flux is given by
'It = t;.. , 'AA% ).
Substituting Into Equation (30),
Su + . (31)
Since •" = __ this may also be written as
IL r -I4 (32)
In other words, an increase in static pressure reduces the momenuum fluxand vice versa.
THE GENERAL MOMENTUM SOLUTION
Consider the control boundary in AA and BB in Figure 24. The horizontalpressure forces on this boundary must equal the momentum flux In the system.
A4.
e~,&~ 1)416 s ;-w (33)
The mass flow is given by ,.a.
3$
"Z C M (35)
and 4 A'g.. (36)
*.1 , (37)
+ for uniform flow. (38)
Thus, as indicated in Figure 25, a reduction in pressure results in in increasein momentum flux, but the change is a little larger than that calculated by assum-ing conservation of total hed. Becasase we use the dimension 9, in Equation(33) the above analysis can only apply for accelerating flow.
We may summarize the results as follows:
"When there is no means of mechanioljly reacting aforce in the direction of interest, momentum (ratherthan total head) is conserved in an invisold Jet."
That is j A-10 4i - J 1 *%! constant. (39)
Thusthe energy (or total head) of the flow will diminish If it is slowed down(diffusion) and increase f it is adelerated. The principle of conservation ofenergy is not violated by this, bwe-uoe we have an essentially unbalanced system.
For the constrained flow case of '1pre 26(a), the ,nmaentum Iblance is
Z36+- 2 6+a. -ZJ lu: A e" (40)
37
2.50 DIFFUSING FLOW ACCELERATING FLOW
2.0
1.0 0
0.5
E(qaatton (731
THE "BORDA-CARNOT" LOSS
-1.0 -0.5 0 +0.5 1.0 1.5
STATIC PRESfJRE PARAMETER
Figure 25. Change in Momentum Flux Through a StaticPressure Change for One-Dimensional Flow.
38
or -,
-40S.(41)
(a) Flow constrained b-y solid (b) 'Unconstrained flow tbhroqjia~wills Static 26ressu -eM
Figure 26. Constrained and Unconstrained Flow Througha static Pressuare Change.
AE indicated in Figure 27, the Ihtra on the left land side of XLpation (41) canhave any value bewn zero O (,I, - S (At, -Af). Itf~t i ero, wehave thecase of no applied mechanical forod and hqAatou (41) is identical with Bquatlon(39).
Referring to Figure 27, this implies that -& )follows the line (A - B-C)In other words, a contra~ction does not take place until after the pressurechange has occurred, Figure 26(b), while an expaision occurs before a pressureincrease.
The second case is certainly true of a sudden duct expansion. However, It isnot clear that such a situation can exist for a reduction of static pressure.
39
1' ' "'4Il A
Figure 27. The Integral A4
(a) Underfed Jet (b) Balanced Jet
Figure 28. Two Jet Flow Conditions.
We mght imagine the condition of Figure 24 to exist for an annular jet. Indeed,most theories "sutms
~5 ~ a constant
rather than
~ - constant.
However, a closer examination of the jet flow indicates (Figure 28) the local
jet curvature itself provides a mechanism for providing the force)
which we have seen to be essential to the conservation of total head.
40
We shall, therefore, continue to assume that when an annular jet acceleratesto ambient static pressure, total head (and, therefore, power) is conserved.
BORDA-CARNOT DIFFUSION IN TWO-DIMENSIONAL FLOW
--- JoI
It.. _____ I AI'|
, I ., .. .+,: . .. . •'
CONTROLVOLUME
Figure 29. A Sudden Duct Expansion.
In this section we consider more formally the simple case of a sudden ductexpansion, which was treated by Borda and Caruot for one-dimensional flow.Following the general aonsiderations introduced In tho previous section, weassume that the expansion of the flow boundary taims place before the pressurerise, as indicated In rilgures 29 and 30.
Using the control boundaries shown in Figure 29, the sum of the pressureforces is
A,('I- At.) (42)
acting positively against the flow direction. (The static pressure at stations(1) and (2) must be constant across the duct, because the flow is assumed tobe parallel.)
41
I
•1,1, 7-3,A
#2
--- FREE BOUNDARY
fr-I.
TYP.
Figure 30. Variables in a Sudden Diffusion.42
The momenaum flux through the control volume is
A. - p ,dA,. (43)0
Equating (42) and (43),
• •�dAzL. (44)
For continuity of mass flow,
o(, ( 45)
The mean velocity is defined as
jAn (46)
Aand we will define the momentum flux also in terms of 44,A,
That is, *044
(47)
where • Av
4(48).
43
Thus, Equations (44) and (45) now become
A~A A-~a) reAit4X4 - ?~ . (49)
CA14AI r At'" (50)
We wish to determine the loss of energy occasioned by a sudden diffusion.Power is defined by
cc A , (51)'3
II+ (52)
where k' Aft
A.- A (53)
Thius,we are interested in the ratio
r. AliaA&- *fA :~
.L "SA .~L *A,4 '4 (54)
From (49) and (50*,
Al 4to + ).AeA . - eA' )QX(A /As)4 XIA64, (A/A)z) . (55)
Substituting in (54).paA)- /A A' ,=
___-_ _, _ _ _, _ _ x, ,__ _,_ _ -__ , (56)
44
A convenient nondlmensionalization is
A (57)
Using this, Equation (56) becomes
'Y a* +. v 4.- 2 A A y>-A/i [-(A~4' 1 ( 8
It is obviously convenient to assume similar velocity profiles at stations (1)and (2), so that4 = -h and N, = N j . We should note that this is onlyan assumption of convenience, however, and that a rapid diffusion can be ex-pected to alter the profile. We would intuitively expect that a un *rm velocityat station (1) would become parabolic at station (2), for example. However,the similarity assumption can be expected to give at least a rough indicationof the effects of profile.
Equation (58) becomes
6T a 'P&- (,_, ,)[(i + ,4,, _ , ] (59)
6• .,..,
It is often more convenient to write the deaonimator In the form
from Equation (52). Thus,
For uniform flow,-f = =1.0, and
45
or A ,-. (I- (61) I
In other words, as JS --0 0 , all the kinetic energy in the flow is lost.
Equation (61) is plotted in Figure 31.
Also in uniform flow
so that
Thus,the total head loss
- . , AV1)L (62)
But
• ". ---. - I " ('-A" A-)'('- gP ). (63)A P,
(63)
,46
1.0
0..lit
0.6
90.4
0.2
R 0.2v
0 1 2 3 4 5
DIFFUSON AREA RATIO As/A,
Figure 31. Power Lost in a kdden Diffusion, fora Uniform Velocity Profile.
47
VALUES OF THE SHAPE FACTORS X AND
• IQ INCREASING
Figure 32. A Family of Radial Velocity Ditributionsin a Circular Duct.
We will consider the case of a circular duct with a velocity distribution of thetype
.,4 (64)
Obviouslythis family tends to a uniform distribution as i.-e and gives atriangular distribution where f. = 1.0.
Now the area A - w'• A., "
Af4AA
48
,0 )
(65)
The momentum integral is
-~~A A0jj - ]1f ,A. 4 JdAt
"= o A - 4 4 ._..- 1 (66)
Thus,from Equation (48)
- (67)
Note thatasw.1.--PO , -- 1.0. Forw'.. =1.0, > =1.5.
The power integral isA^ 4' a -M'ACf-f jA fs#-7i +~-~J
3 •- g. -
a -- • (68)
Thus, from Equation (53)
4*9. * (69)
49
As*- -*oo ,• %P- 1.0. Forw- = 1,0• = 3.7.
Equations (67) and (69) are plotted in Figure 33, and the resultant power lossparameter, from Equation (60), in Figure 34.
THE LIMITING VELOCITY CHANGE
From Equations (49) and (50),
or
A, A,,' I J"L > 4 :
f A \ A )(711
For similar profils )"s. "
an equation which Is plotted In Figuzre 35.
If we write Equation (71) as4, x
-E (7 4:
50
UNIFORM VELOCITY TRIANGULAR VELOCITYDISTRIBUTION DISTRIBUTION
0 ,0
0
0 0.2 0.4 0.6 0.8 1.0
DISTRIBUTION SHAPE FACTOR A
Figure 33. Variation of the Shape Integrals )X and kWith the Shape Exponent e# for the PowerLaw Family of Velocity Distributions.
51
,s/AS =2.5
1.4 SUNIFORM VELOCITY A 1
DISTRIBUTION -
1.2
1.0
%0.8_____
0.6__ _ _ _
0.4
0.2 /
00 0.2 0.4 0.6 0.8 1.0
DISTRIBUTION SHAPE FACTOR t//*
Figure 34. Power Lost in a Sudden Diffusion, as aFunction of the Velocity Distribution Profile.
52
1.0
•0.8
S0.6
0• 000541
0.4
S/0.2
0 0.2 0.4 0.6 0.8 1.0
PRESSmRE RISE PARAMETLR
Figure 35. Variation ,f Velocity Ratio 4//, , Withthe Pressure Rise Parameter Cp , forSimilar Profiles.
53
and differentiate with respect to , then equating to zero gives
Substituting in (74),.
(76)
Thus we see from (75) and (76) that, so long as the initial and final velocity
profiles are similar (At = A ), Cp,. = 4 at a mean velocity ratio
Thus increasing the flow distortion increases the pressure rise coefficient,although at the expense of increasing the associated energy loss.
Since X increases in value with increasing deviation from uniform flow,however, we can expect At > X,, when the profiles are not similar. Thus,distortion of the flow profile dur diffusion reduces the maximum obtainablepressure rise.
The corresponding 'otal pressure loss for uniform flow is plotted against CpIn Figure 36.
The critical velocity ratio of tM, is extremely interesting, because thishappens to be appropriately the limit for all diffusion processes. Ackeret'scriterion of
Cp > 0'7 t- o. v (77)
for stall ha been confirmed experimentally by Schlichting5 . If 7 is a diffusionefficiency, we have from Bernoulli, A, As
4A A
54
•"1.0
CLO" 0"•.• "
0.6
0.4
0.2
0 * 2 _____ _____
0 0.1 0.2 0.3 0.4 0.5
PRESSURE RsE PARAMETER
Figure 36. Variation of Total Pressure Loss With StaticPressure Rise, for the Case of UniformVelocity Distribution.
05
S(78)
For uniform flow, and 1. 0 (no diffusion lossý a value of " y =, --
gives Cp = 0. 75, a value which is in excellent agreement with Ackeret'scriterion.
Lieblein6 has shown that flow separation occurs on compressor blades whenthe ratio ot the peak velocity to the trailing edge velocity exceeds 2. 0.
Other examples of this limitation on diffusion have been cited by Senoo 7 whopoints out that all conical diffuser data combine to show a peak pressurerecovery coefficient in the range 0. 7 - 0. 85, corresponding to velocity ratiosof about 0. 55 to 0.4. He also shows that, using numerical methods of determin-ing the stability of a turbulent boundary layer, a uniform diffusion causesseparation at a velocity ratio of around 0.6.
Finally, Goldschmied8 has shown that stall can be correlated by the equation
Co so g0o z (79)
whe,'e 2; is the skin friction stress at the point of maximum velocity. SinceScannot exceed. 0048, Co, cannot exceed 0. 86 under the most favorable
conditions (& = 5000) and will be loss for the higher Reynolds numbers en-countered in most practical problems.
Thus.all the experimental evidence seems to point toward the essential correct-ness of Eqiatlons (75) and (76), even though we cannot at present see clearlywhy this should be so. There is obviously room for a great deal more researchIn this regard.
So far as the prse"t program Is concerned, we are principally interested inEquation (63), which can obviously be expressed as
A )P, (80)
56
AP (81
Or, sinle AA-C -
When dealing with a curvilinear flow field which has a velocity gradient acrossthe direction of flow, we shall assume that Equation (81) applies along individualstreamlines. This implies that, even in the middle of a let, diffusion takesplace at constant static pressure (Figure 29 on an elemental scale) and thisassumption is impossible to justify, at the moment, other than by the resultsof analysis based upon it.
57
Chapter ThreeTHEE QUATION FOR CURVnIT OV IMNINAFO
THE EQUATION OF MOTION
(a) Curved flow e et
(C) Dhimenai~g ofazel m tF l g u 3 7. T h e x ls mnl* t i G eo %o mtzy O f th e C u rv e dFlow Field.
Fn pro unit~ 37, ter eamatf Pr eagur force acting (lnward) Or' an eleme ntiF~
(8~ 2)tlegh
Taking only first-order differentials, this simplifies to
Ft d (83)
The mass per unit length of the element is
e-- C" (84)
so that the centrifugal force is
-1-- - (85)V-
Since the centrifugal force must be balanced by the pressure force, we canequate (83) and (85)
S= e' . (86)
The local velocity is related to the total head at radius " by Bernoulli'sequation %
44
Substituting for i. in Equation (86),
d (87)
where can be any funotli of -r
In order to solve this equation we need to specify one boundary condition;usually the static pressure at the Vp, or R boundary. For exsznple,specifying ambient static pressure at the O botindary would give the an-nular jet solution; specifying ambient pressure at the R boundary wouldcorrespond to Coanda flow around a curved surface.
59
It is convenient to use "gauge" pressures (that Is, pressures measured withrespect to some datum pressure, usually the ambient value), the definitionsbeing
AP P(88)
It follows that Oi •. Thus Equation (87) becomes
Eai (89)
A LINEAR TRANSFORMATION TO JET ORDINATES
Since the total pressure is specified with respect to the jet ordinate ( ! inFigure 37) it is convenient to transform Equation (89) to this variable. Sucha transformation also enables us to introduce the concept of a jet which has alMnear variation of curvature across its width, by writing
-P * -t -+ 2 (90)
where f] 1.0 for free vortex flow.7 = 0 for the constant radius assumption
used to derive "exponential theory".
In general, 04.7 4 1.0 for an annular Jet, for example.
Note that V - 1.0 In Zq•Wtion (89)% despite the transformation. It nowbecomes74
& 4 t-~ (1
The general solution is the familiar formula for a first order linear equation:
60
4- 1.- - r •/( $(92)
ti~cosan < cnzr if/ 0. whn5=- ,o ous.Thooa
I .. --
- r. 1 93)
%•!.-'titu im ii (92.1 ..
tilt, Cons•tant bei~ng zero if •4,=0 when =0, of course. The total
t:Ltic :pressure rise across the jet is
,NINEAR TRANSFORMATIONS
:'.l•iough not gerriiane ., the problems of the present program, it is ofnttcrist to note that nonlinear transformations of Ecuation (89) have value
ccrtain cases, one example being depicted in Figure 38.
61
Figure 38. A Flow Which Has Infinite Radius of Curvatureat the Center Line.
F-ir this case, for example, we might reasonably use the transformation
A-t. (96)
Thus Equation (89) becomes
A (A-" (97)
62
Chapter Four
INVISCID FLOW ANNULAR JET THEORY FOR CONSTANT TOTAL HEAD
In this clhpter we shall apply the curvilinear jet theory first derived to theannular jet problem. Of all the common curved jet flows this has been studiedmost extensively, both theoretically and experimentally, and is therefore avery logical starting point.
As noted in the introduction, most theoretical analyses to date have been basedon momenti"m. equations. The well-known exception is Strand's14 -potentialflow solutiun, which is known to give erroneous results at low ground clearanceheights.
When an annular jet is close to ground plane, it approaches, in the lirmit, thecase of a plenum chamber. The plenum chamber problem can be regarded asequivalent to a jet issuing from a slot in the wall of a large vessel, the groundplane replacing the axis of symmetry.
An extremely complete fnily of solutions for finite plenum chambers hasbeen obtained by Gabbay , as part of a more general investigation, and theappropriate results are summarized in Figure 39. Since these results passthrough the already known points for 0 - + 900, 00 and - 900, and sincethese isolated cases are known to agree well with experiment, it is reason-able to feel a high level of confidence in all Gabbay's results.
In passing, it is of interest tc note that the existence of these results nowenables us to work out definitive solutions to the problem of the optimumwall angle for a plenum chamber GUM.
GENERAL MOMENTUM BALANC• COKIDERATUONB
A Thin Jet Solution (Pinnes' Oeometrvd
When L/R 1.0, we may postulate the geometry of Figure 40 for an annularjet. The assumption of constant radius of curvature along any streamlinefollows, of course, from the assumption of constant cushion and ambientstatic pressure.
We refer all pressures to ambient by the ( A ) notation. Considering thecurved part of the jet (AABB), since the external pressure forces must beequal to the change in momentum, we resolve horizontally and vertically.
63
f
1.0
0.65
0.
0..2
0
-900 -600 0 500 900
WALL ANGLE IN DEGREES
Figure 39. Potential Flow Solutions by Gabbay Appliedto the Plenum Chamber Problem.
64
U , -
- -IIi • -
I '66
For the vertical resolution,
- Ce (98)
For the horizontal resolution,
4-d - 3S 4 ..1ea- A sLZ1 Q
or, substituting from Equation (98) for j%
Considering now the momentum balance upon the accelerating flow section
(BBCC), the horizontal resolution gives
ts h(10r)-T "
It should be noe that no asssumptonn have had to be made for the form of the
static pressure distribution across the jet or for the total pressure variation.
Thus Equation (101) should hold for any theory.
The present analysis does not completely define the effective jet periphery 4,of course, except in the two-dimensiona flow case. However, a three-
dimensional momentum balance is easy to do for a given three-
dimensional planform, using the concepts introduced above.
66
A Thick Jet Solution
As the jet nozzle approaches closer to the ground, we cannot expect the region(BBCC) in Figure 40 to be physically separated from the curved jet flow. Rather,as Strand has shown in his legant potential flow solution, the acceleration toambient conditions will take place during the curved flow regime, and the jetshape will resemble Figure 41.
Equation (98) is obviously still applicable to this case, so an expression de-fining the nozzle pressure integral is available. For the horizontal resolutionwe have
46 = " (102)
e.
9- bstituting Equation (98) for Jd .we find that Equation (101) still applies.
The Variation of Curvature Across a Thick Jet
The outer (cushion boundary) radius is always defined by
L. R "4" .0,;, ) -- (103)
on the assumption that the radius along any streamline is constan.
For the geometry of Figure 40
0o - (104)
and the flow pattern is oompoetely defned. Note however, that In previouswork the (t**4-0) term in l•qmttnh (103) has been omitted.
For the thick-jet can of FIgure 41, we must mai. an additional asamption;to whit, that the ambient boundary streamline is tangential to the nozzle wall.Then from Figure 41
I? a . +a ) r, *,# t:.ort
67
L 4g -- '
I ",• • "
J / , )
"• I '
68
if +1. 7 • , as discussed in the previous chapter,
and 1z ( - -t (106)
But, from Equation (105),
The ratio is a discharge coefficient, of course. As
0~ 0 6
The Height Parameter '/V
We shall find that the fundamental parameter which defines the characteristicsof an amular Jet is
,-is_ O=bo ,44 dk MathI4Y ams of ouayreI of tho iner Str~eamine
From Equation (103) this is gives by
% 4 (108)
Previously, most writers bve used th umeter
! �.(I 4 •, s e) (109)
69
which is evidently erroneous, for 0>0, from the geometry of Figures40 and 41.
Three-Dimensional Effects
The principal geometric effect of three-dimensional flow is to modify the jetcurvature and hence the cushion pressure. From Figure 41 it is obvious thatthe radius R will not be influenced, whatever the planform. When the jet iscurved in the third dimension, however, as in a circular planform GEM, forexample, its effective periphery increases as it moves away from the vehicle,so that its thickness ( -t, ) on the ground plane is accordingly reduced, leadingto an increase in the ambient side radius "r. .
It irs important to realize that this effect applies only to annular jets which donot have straight nozzles in planform. The jets of a rectangular GEM wouldbehave (on this postulation) exactly as if they were two-dimensional, the"three-dimensional effects" being confined to their junctions at the corners.There is some experimental evidence 1 9 to indicate that, in such a case, theinteraction of two peripheral jets at a corner gives an increase in cushionpressure rather than a loss.
As the most common example of a curved jet, the circular planform ilus-
trated in Figure A2 is considered here.
Figure 42. Geometry of a Circular GEM.
The effective ciroumferemo C' of the Jet is given by evaluating the area of thefrustum of the right ciroular oone made by the Jet exit.
70
C-i:L
Ca ii(110)The jet radius on the cushion side is unaffected by the planform. The radiusdefined by the V; arc becoming parallel to the ground is
Thus the effective pe•iphery is
The analysis then proceeds as before. However, in now defined an
. .• x ,-I,
or * I * (114)
The limit case of a very thia Jt Is provided, of ourms, by asmaming
-I.l 1.0.
SOME 2ENUAL UZIMTK woM TEE FLOW PARAMETEM OF AN ANNULARJET
in the most general oase, the toW bead wriatiom across a jet is a function onlyof the distance ( ) across t jet.
(1
71
The local static pressure A+ is also a function of It is thereforepossible to define completely jeneral relationships for the various performanoeparameters of interest, in terms of these two variables.
The Local Jet Velocity '4"
In the nozzle cwt plane, the local velocity is
(116)
The Jet Mass Flow 1EV.j
e C
The Nozzle Momentum Flux 53
2. - - (118)ECA
Total Noasl Force
sI . . , (119)
It
72
Jet Power
Power is defined at the nozzle
,.) - .0 IAP d,\
-ip OR-A P ( (120)
Total Lift
Total lift jet lift + cushion lift
- r FC.,&* A* (121)
in inviscid flow.
SOLUTION OF THE CURVED JEt EQUATION FOR CONSTANT TOTALPRESSURE
We come now to the colution of Eqation (94) fdr the case
, .constant=,,(Bay).
This !ives
+ 75) 0 .? *7 ) ), (122)
the constant of Integration vanishing beomau of tim boundary condtion A f 0at • -0. Integrating(122),we obtain
7(123)
73
and, of course, the cushion preewre to
LAcWI!Wq Velocity
Jet Kass Flw j
C (St6pi) 6%I
The Notlk MamommFlm r
Tota Jet Force F,.,
74
2CA 09•A P
S , the preceding section;
J. let Curvature Parameter
Li,,h jet has accelerated to ambient, its thickness is
N11W", for continuity
t~i"
P -L -(129)
There l rc two possible assumptions for calculating . If total head isS)Ilse!rved, then
* ~b, '~- ~(130)
rumi, Equ~ation (126).
I L'al I momentum is conserved, then is not necessarily constant across* Th, conservation of total momentum assumption only states that
J. C f, • • J,,. (131)
,Th.precsecnt analysis we will assume 1 ,= a constant, while notingilmintations of this assumption. ThusEquation (131) becomes
75..
-,5 . ) (132)
Substituting Equations (126) and (l1.8)
A x I -t? (133)* _kr, ,1" tŽ
2-1--_
The jet thickness ratio AiA is also defined by the jet gometry. If we assumethat the inner and outer boundaries are circular arcs, then from Figure 41
(134)
By cross-plotting Equation (134) with Equations (130) or (133),we can obtainsolutions for 7 in terms of 'AW/W . This has been done in Figure 43, assum-ing conservation of total hued. The resulting values of V? and ir U.Ceplotted in FigUre. 44 and 45. The agreement with Strand's potential flowsolution is notable In F/Iure 45.
MomeNtum Balance
From Equation (136), oomwning oonservation of total head
'. -; -i t -IV .- • * L'- •'P•" (135)
From EquatIon (101)
(136)
76
T .0 Sim 0
1.0
9.05.0
0.8 2.0
01 1.0
F-4 0.60 / / 0.5
0.30.4 0.25
0.2
0,- I0. 1
0 C.2 0.4 0.6 0.8 1.0
JET CURVATURE PARAMETER 1
Figure 43. Cross-Plot of Equations (130) and (134) ToDetermine Values for the Jet CurvatureParameter ¶ . (Solution for Conservationof Total Head.)
77
1.0
0. 8NS-
0 0.6 =0
[-4
i.0.4
0.2
0 0.5 1.0 1.5 2.0 2.5
HUGHT PARAMETER t/9
TIpr 4C, Vaiatift of Jet Curvature Parameter 91W*h Lb eight Parmeter E/*.
78
00
E-4 N -
______ ~ lbo __ _
-raC
0
cq
~II II ~ C§co
to
- .,9OLLWV SUZNNULL 1ZP
79
Comparing this with Equation (117) we have
(Momentum Balance)(Curved Jet Theory)
S( (13b
-Equation (138) is plotted in Figure 46. Although the momentum balance breaksdown as -P O, the new formulation is obviously much more satisfactorythan previous theories. It should also be noted that the divergence from 0 = 1. (as R --s 0 may not indicate a departure from balance, but only a departurefrom the assumption of circular curvature.
Other Results
Knowing the variation of w with , we can calculate the values of theother quantities derived earlier in this chapter.
In Figure 47 the cushion pressure parameter A1114'p. behaves just as we wouldeect., iallhng between the free-vortex and expofiential theory results.
The discharge coefficient plotted in Figure 48 Is particularly interesting, be-cause, in onatrast to previous results, it obviously agrees well with the knownpotential flow solutions for a plemam chamber. For the case G = 0 on thecurve obviously fairs well into the point CS = 0.61 which is established for ajet issuaing fromt an orifioe in a plane wall, while the O = 900 curve would ob-viously fair into the Borda mouthpiece solution of C4 =0. 5.
It will be recalled that a major criticism of itrand's potential flow solutionwas the limit SC -o 0.5 as A...* 0 in contrast to the known result Ci, = 0.61.
A more detailed comparison of the discharge coefficient predicted by thevarious theories ih liven in Figures 49 - 51, which also include an analogoussolution by (;bayiI8
so
1.4
1.2
EXPONENTIAL THEORY04 • ----.- _i • . •,..,'-
_______ ~PRESENT THEORY ____0 1.0
z
Sr ..,FREE-VORTEX FLOWW 08 VI -".,. Al /~-11
00
rz o.4
442 4A6 8 10
HEIGHT PARAMEER /g
Figure 46. Momentum Batanog as a Function ofthe Heigit Parametr A4.
81
1.0
PRESENT THEORY
0.8 600
-.- FREE-VORTEX FLOW
-• '\, I4C0 6 'EXPONENTIAL THEORY -
o.0~0.4
j0.2
0 2 4 6 8 10
HIMGHT PARAMETER *4
Figure 47. Variation of Cushion Pressure With theHeight Parameter R/t.
82
0.7
GABBAY1 8 SOLUTIONS
0.6_____ _
0.5 \
0.4
S0.3
S0.2
0.1 0
0 2 4 6 10HEIGHT PARAMETER
Figure 48. Discharge Coefficient a . a Function ofthe Height Parameter Qe/It.
83
0.7
0. 6
GABBAY-4_.>
N STRAND0.5
• fSTRIAND
S0.4
44
0
0.3
0.2
0.1
0 1 2 3 4
HEIGHT PARAMETER -A/96
Figure 49. Comparison of Theoretical Discharge
Coefficients for 0 = O.
84
0.7
0.6 GABB''- 300,
STRAND
S0.3
z PAYNE0.4 ,.
8 STRAND
NGABBAY
o.2
0. 1 _ _ _ _
0 1 2 3 4 5
HEIGHT PARAMETER -11t
Figure 50. Comparison of Theoretical DischargeCoefficients for E = 300.
85
0.7 F I ,
GABBAY
0.6 __00
t .).44 \
-WPAYNE,
41.3
0.1
I-i _______/___________
1 2 3 4
MUGHT PARAMMURt./t
II. •m of Theorwtal DischargeS.for0 =60o.
S
Gabbay's problem involves a straight wall as the inner (cushion side) boundaryinstead of a constant static pressure streamline. Since the static pressurealong his boundary is not necessarily constant. his result approximatec3 to theannular jet case only in the limit ,/t - 0. But at large values of the jet angle0 rhe approximation wi]l presumably be valid at finite small values of
". ste that, in gene: ;i, the pr-sent theory agrees with Strand's for AA > 1and tcnds tot=ds Gubbay's for 0. 5 "< . . At values of Ao4 4 0. 5, thesolution breaks do',n because the implied geometrical a.r;sumptions are nolonger tenable; but the e: 1 rapolktior. from higher values points quite convinc-iri"v towards Gabbay's limits for A4,• 0.
Srractical hI - rw:ge of izrtere t is 0. 5 - V<2 or 3. 1 he elegant.k.d sophistica0cd,4i-v ' 'al f-ow frei-nMc,:ts of Gahbal"and Strand apply respec-tix'e, t,)eiow avrA above itA.s rangn. 7"ithin the range, this relatively simple.i,,or•. If this& repo-4 is af•-Va•-f-.W .-- ca
,.\f i w is a r.•y n,.. ap'licablc than either.
J .ould Labien to add ti.at Gahbay va.'? not considering the annular jet prob-,i in 1-Is analysi, and wie mr kitemrJsd his svlutton to apply to it.
fit nozzle force parameter plotted in Figure 02 is noteworthy because thepr. . c lift termn i:; gcn-erally Ig-;aored in the c.alc-lation of total lift, yet is.:cet to make an import.rnt -ontribution tt low grouad clearance heights.
Sor, of the calculated two-dimer-tontal solutions are gfren :umerocally in
The same procedures apply to the cak-ulat.onof Jet chrafteristics in thethrce-dinierslonal ease, using Eqqation '114).
A, an exiaimwe, -&e calculate the valuex of Y7 and * . fnr a circula plan-t,,-m GEM, for the ase =0 0. 1165. The resulto are plotted InFigures T,3 and 54, and we fee that, although therv is a significant varltioni,* 7 ( and therefore ke/k), the cushion pressure rmmafbs a'1monrt completelyunadfecled by the change of planform.
£11," C(`SJlON PRESSURE PARAMETE,
e.'c shall see later Luat Ime conveuitional cushion pres&cre pa-amit(or
87
1.0 1
0.8 "000
i 0oo0.4 CONTRIBUTION OF THlE
O NOZZLE STATIC PRESSUREDISTRIBUTION
0.2
0 2 4 6 8 10
HEIGHT PARAMETER ft
Figure 52. Total Nozzle Force, Measured Parallel
to the Nozzle Axis. as a Function of theHeight Parameter J/tt
88
0C4
Vý 40 0 0 0 0 0000000000 L QC9toto00
00000000 000t tt 0000 qel00000-
m m00 m n t I-" 0 (M m o 0 wJ4totfý -P0 0 m~mmmlI- L~f0t~~ -'Mý
0 C
CID oo.. 00 Chl C43b O t- OD
89
1.0
10.8
TWO-DIMENSIONAL SOLUTION
0.6
0.4
CIRCULAR PLANFORMV. = 0. 1135
0 0.5 1.0 1.5 2.0 2.5
HEm" RATIO Vi
Pip" 53. Comparison of Disoharge Coefficient RatioAb/f for Two-Dimensional Wd CircularPlanfmm GZMs (I =00).
90
1.0
0.8
it TWO-DIMENSIONAL SOLUTION0.6
0.4
CIRCULAR PLANFORMz t/.P 0 . 1125
0.2
0 1 4 6 S 10
Fipro 54. iotd ol Pl•.m cm ftb Cnmcm
91
•,. = cushion pressure
A Pi mean jet total pressure
is not a very meaningful index of annular jet performance, nor is it a goodparameter for correlating test data.
The fundamental performance parameter in hover iE the ratio of the total liftdeveloped to the power lost in the jet, but this introduces extraneous variablesrelated to the planform and geometry of the vehicle. A closely related indexis the ratio of the cushion pressure to the jet power, however, and this seemsto be the most useful parameter for general use.
From Equation (120) the general expression for jet power is seen to be
-!- t - ý )& .) (139
Since VlL P. jO L
we may expect that
1Kconstant.
We can formalize this result by oonsldering an Ideal (very deep) plenum chiamberat an edge clearance height of A& . The mass flow out is
W; - e5- (140,
r (141
The obvious non-dimnsim alizaton is" "C- (142
92
A6can thus be calculated for any GEM confiuraton. provided thatandAIL are known. Alhuh is the reciprocal of thi is aneffective discharge coefficient. In the general case, it will differ from theactual discharge coefficient, which is defined as
CI, * (143)
Naturally, although we have derived this relationship from plenum chambertheory, it can be applied to an annular jet. Indeed, at very low edge clearanceheights ( -4.4 O),there is no difference between the two configurations. Aslift increases, the annular jet may be regarded as a special case of the plemm,where dLcting is employed to reduce the discharge coefficient.
The General Expression for
From Equations (139) and (142),
Substituting for and Wo M s (H) OWd ),
A+ - .2 (145
An optimum total bead dldOu M b WIamwl bern which maule- hi expresson.
The Cousta Nt Roda Radi
If i a 0omt"tN", it vani/shes &om the suo, sad
C I Or(*4 14
93
that In, i (146)
- (147)-.[' -••÷C,
The Solution for Free-Vortex Flow ( 10 = .0
For the case b7 = 1.0,
A•5 _ (-.,'•). P, CL) (148)
Substituting in Equation (144),this reduces to
IL ' I)(149)
which can be written as
if I t (1 5 0 )
The loautm for 3•Mg 4s % mo•r k Flow ( - 0)
In th oam*itam be shmw tUat
4 . (151'
94
GENERAL SOLUTIONS FOR FREE-VORTEX FLOW WITH CONSTANT TOTALPRESSURE
For the special case of free-vortex flow, the jet curvature parameter isS= 1. 0. Thusfrom Equation (94),
f(152)-z.
"- •-"-" 5 ("• " )"(153)
Calculation of Mass Flow ni
From Equation (117),
-__-- .. ,-•÷)-. •¢
,. (154)
* t~~3 -~ *e~ji+ ; (155)
writing
and I_ (1AO)
Calculation of the Lool VlootY
.An (157)
S- 4. ,(15.)
95
Note thatasNA-w 1. 0, • 0) .
The Nozzle Mouientum Flux Jj
Since .- -- . .2 P
2Ctq 1145S
The Total Jet Force F
From Equation (119),
ZrrI+
(16o)
The Jet Force
From Zqiom (117) and (12@Vt is obvious ta the Jet power parameter isthe same a the maw flow prmeter.
The momentum flux to ambient, mumming conservation of total head, is
96
Aj.(Momen'im -hiory; 0-*A IT4(12A + ,, ( C u r v e d J -at T i t ,"o r y ) -t -(
?.ANERAL SOLUTIONS FOR CONrSTANT RADWS FLOW (EXPONENTIALTHEORY) WITH CONSTANT TOTAL PRESSURE
The exponential theory is concerned with the solution to Equation (94) whenS= 0. W e then have, from Equation (91),
._ -~ [z•f l • •(163)
Wheno o; .'.k 0
WMen a Constant , atim (164) beomes
, '6 -- e(165)
and. of oourse, the cushion pressu-e is
AN ~ - 4
(19797
The Local Jet Velocity V 1 f
Since -j
C. (167)
The Jet Mass Flow li
Since C
C)'7 Ar
(168)
The Naszl. Momntmb Flux
-- q
/ 4L
- I-. •(169)
Tho TOW Jet oe
"" •.•4p. 2&-• ,.',, . 4e aVI . (170)
98
The Jet Power
From Equation (120),
This expression is the same as for the mass flow, of course, so that
' (171)
Momentum Balance
The momentum flux to ambient, assuming conservation of total head, is
. -. -I?
From Equation (101),
.- )- (172)
Comparing this with EVUltio (16)0
(Curved Jet Theory) (17)
99
SOME EXPERIMENTAL REJLTS AT CONSTANT TOTAL PRESMYRE
The Anmtlar Jet Test Rix
The annular Jet test rig shown in Figure 55 is based on a concept by Dr. HarveyChaplin of the David Taylor Model Basin. The novelty lies chiefly in the use ofan air supply at room ambient conditions %spirated through the test section toartificially low "ambient" conditions created by a centrifugal blower.
This arrangement has the ad'antage that, since the supply air is initially stii,the distribution of total pressure at the no7 " very uniform and losses anddistortions arising from boundary layer effects are minimized. The use of aconstant static pressure streamline intake assists in the creation of these de-sirable flow characteristics.
The appearance of the rig, and of the nozzle geometry used for the presentseries of measurements, is shown in the photograph, Figure 55. The internaldimensions and location of the measurement stations are shown in Figure 56.The rig is practically two-dimensional, being contained between the sideboards11.5 inches apart. All the measurements were made in the central plane andquantities such as mass flow, volume flow, and power are referred to a unit wiclh( C"' = I foot). Thus, flow volume appears as cubic feet per second per footand power as foot pounds per second per foot.
The Expzeriment
The principal plqperinntal subject is, of course, the behavior of the air atthe nozzle exit. lise air ooditions prior to entering the nozzle were knownabsolutely, being the amblent ruom conditions. The principal measurementswere at the cutlet, where total and static pressures and the angle of flow wereobtained. A flthor series of meswurementd were made a little lower down-stream, traversing the jet ad deep into the cushion to examine the behaviorof the cushion air mass.
Total and static pressures in the Jet were sensed with a probe which detectedalso the local drection of the flow. The sensing head of thls type of probe iswedge shaped, with the total pressure tap in the thin leading edge and a statictap in each slant face. The observed static pressures equalize at the truestatic when the wedge is pointing upstream, subject to a correction for errorin manufacture and for pitch angle of the flow. The probe was inserted throughthe side of the rigand the traverse was obtained by swinging the probe about
100
-00
101
DETAIL OF PROBE
"HIGH" CUSHION
REMOVABLE
"LOW'r 56 nera ienin o nuarJtTetRg
BO102
the point of penetration of the side wall. Thus, we obtained the angle of theflow in the central plane, as well as the pressures, and were able to correctfor the pitch component due to the probe angle relative to the central plane byusing the calibration. (Actually, this correction was very small at all positions.)Pressures were measured on inclined manometers. The readings were alwaysconsistent and repeatable and are believed to be free from random error within± 1 percent of the maxima, including transcription and computation errors. Sys-tematic errors, if any, are unknown and are inherent in the method.
The pressure and velocity at the nozzle throat and outlet are shown in Figures57 and 58. The experimental data are given in Table 2. These have been re-duoed from manometer readings to pressures in inches of water and velocity infeet per second. No smoothing has been applied, either by alteration or byomission. Values for the flow volume and flow power were derived by numeri-cal integration of these data.
Flow at the Throat (Figure 57)
The total rressure at thi throat was virtually zero (relative to room ambient)because of the negligibly jmaU loss incurred by the flow in the entry. Thestatic pressure across the throat was nearly uniform with a rml-ght linear in-crease outward fron-m the cushion side. The mean velocity was .' 3 feet persecond, and the volume flow was 9. 00 cubic feet per second per foot. Thisvalue is probably slightly low because of the inability of the probe to penetratethe very thin boundary layers and the consequent omission of some small partof the flow from the integration. Similarly, the flow power, at 137 ioot-poundsper foot, may also be low.
Flow at the Nozzle
At the nozzle outlet (Figure 58) the flow pattern in greatly changed. There isa velocity gradient running from a sharp peak value at the nozzle lip down tozero at a point 2.25 inches nearer the cushion. The flow angle is, In the main,normal to the traverse axis, indicating that the air has already turned throughthe nozzle angle (300) and more toward the lip. Numerical integration of thisvelocity across the traverse gives a flow volume of 9.06 cubic feet per secondper foot in the very fair agreement with the throat value.
The static pressure beyond the lip drops sharply to a depression of 2.94 inchesof water. This Is artificial "ambient" created by the inlet to the blower. Thepressure scale in the figure is referred to this datum, giving a cushion pressureof +2.63 inches of water and a supply pressure of +2.94 inches of water. The totalpressure loss across the jet is greatest at the cushion side and diminishes to
103
""CUSHION4.0 1
043.0 1___ A1P 12.94
0
~2.0 100 >
1.0 50
0 02.0 1.5 1.0 0.5 0
DIMTANCE FROM DUCT WALL IN INCHES
Total paeuueAP and static pressureA# arerelative to the artificial ambient pressure at outlet.
Mea velocity - 56.8 ft/sec.Flow volume - 9. 00 ft2 /sec.Flow power = 137.0 ft-lb/sec. ft.
Figure 57. Velocity and Pressure Distribution inthe Throat of the Annular Jet.
104
AXIS OF TRAVERSE FLOW DIRECTION
4.0 1-f~ IITTV~f
w 3.0 94__
2..620
raw
1.0 50
2.0 1.0 0
DISTANCE INWARD FROM NOZZLE LIP IN INCHES
Figure 58. Pressures anI Velocity at Nozzle Exitt*/-1. 1 . - 300 4 -2. o0nches.
105
TABLE 2
EXPERIMENTAL DATA - ANNULAR JET TEST RIG
Location Static Total Flow VelocityInches Pressure Pressure Angle Feet PerInward Inches Inches Degrees SecondFrom Lip Water Water
THROAT
0.15 2.19 2.94 - 57.60.25 2.17 2.94 - 58.50.35 2.18 2.94 - 58.00.45 2.18 2.94 - 58.00.55 2.20 2.94 - 57.30.650.75 2.20 2.94 - 57.30.850.95 2.23 2.94 - 56.21.051.15 2.25 2.94 - 55.41.251.35 2.26 2.94 - 55.01.451.55 2.28 2.94 - 54.21.65 2.28 2.94 - 54.21.75 2.29 2.94 - 53.81.85 2.31 2.94 - 53.01.95 2.32 2.94 - 52.5
NOZZLE
0.14 0.000 0.070 - 16.50.08 0.000 0.070 - 16.50.02 0.040 0.070 - 9.50.04 0.000 2.768 +10 111.00.10 0.590 2.770 - 99.00.16 1.220 2.850 +3 85.00.29 1.640 2.886 +1.5 75.00.43 1.880 2.884 +1.5 67.00.54 2.023 2.883 +6 62.0
106
0.66 2.150 2.877 +6 57.00.79 2.223 2.864 +6 53.50.91 2.310 2.855 +1.5 50.01.04 2.361 2.846 +1.5 46.51.16 2.t25 2.824 -1 42.01.29 2.459 2.824 -1 40.5.43 2.490 2.792 -1 37.0
1.54 2.535 2.757 -1 31.51.66 2.544 2.747 -6 30.01.79 2.570 2.716 -11 25.51.91 2.580 2.694 -18 22.52.04 2.596 2.663 -19 17.02.16 2.598 2.636 -26 13.02.29 2.609 2.609 - 0.02.43 2.615 2.609 -
2.54 2.631 2.631 0.0
NOZZLE EXIT AND CUSHION - LOW CUSHION BOARD
-0.41 0.00 0.10 90 21-0.29 0.00 0.10 63 21-0.16 0.09 2.21 48 97-0.10 0.35 2.62 37 100.5-0.04 0.70 2.62 30 92.5+0.08 1.165 2.82 25 86
0.21 1.48 2.84 23 77.70.33 2.745 2.85 20 700.46 1.925 2.82 12 630.58 2.080 2.829 1 560.71 2.178 2.827 12.5 540.83 2.275 2.816 12.5 490.96 2.332 2.798 12.5 461.08 2.399 2.776 11 411.21 2.426 2.758 8 391.33 2.460 2.743 6.5 361.46 2.494 2.722 2.5 321.58 2.515 2.697 -2.5 281.71 2.529 2.695 -3 271.83 2.542 2.650 -5 221.96 2.550 2.639 -7.5 202.08 2.559 2.639 -10 192.21 2.569 2.612 -20 13.8
107
2.33 2.573 2.590 -22 9.22.46 2.572 2.583 -2. 9.02.58 2.569 2.569 - 02.71 2.569 2.569 - 02.832.9f 2.569 2.569 - 03.083.21 2.569 2.569 - 0
3.33 2.567 2.574 +75 5.63.46 2.567 2.574 +80 5.63.583.71 2.574 2.574 - 03.833.96 2.577 2.577 - 04.084.21 2. 571 2.573 +90 3.0
NOZZLE EXIT AND CUSHION - HIgH CUSHION BOAR')
-0.41 0.00 0.30 90 36-0.29 0.00 0.28 60 35-0.16 0.17 1.25 51 63-0.10 0.30 2.833 40 106-0.04 0.63 2.825 29 99+0.08 1.20 2.833 26 850.21 1.545 3.060 24 820.33 1.870 3.026 19 77.10.46 2.075 3.0(C8 19 640.58 2.207 3.008 17 59.20.71 2.320 2.999 17 54.60.83 2.408 2.968 16.5 49.60.96 2.495 2.920 15 45.71.08 2.538 2.928 15 41.41.21 2.604 2.929 10 37.b1.33 2.604 2.929 10 37.81.46 2.675 2.860 8 28.51.58 2.675 2.860 8 28.51.71 2.719 2.806 4 19.51.83 2.732 2.825 20.21. 6 2.745 2.816 - 17.62.08 2.749 2.780 - 11.6
2.21 2.757 2.771 - 7.8
108
2.46 2.746 2.751 4.?l2.58 2.757 2.7V- 0.02.71 2.762 2.762 10 0.102.832.96 2.758 2.7"1 12z 3.63.083. 2. 2.758 2.776 87 8 93.333.46 2.789 2.789 47 10.03,583.71 2.789 2.789 29 5.13.833.96 2.789 2.789 24 5.14.084.21 2.789 2.789 24 5.1
109
very small values toward the outer edge.
The power loss between the throat .. ad the nozzle is simply the integral ofP.%rd6 across the jet at the nozzle. This was computed as 5.49 foot-poundsper foot, equal to 4.01 percent of the throat power. The total pressure lossis plotted in Figure 59. The power distribution in the jet, like the velocity ofthe flow, is weighted heavily away from the cushion. It ib given by the productof velocity and total pressure referred to the artificial ambient; that is to say,by the product of WO and -%r as they appear in Figure 58. This is also plottedin Figure 59.
The static pressure variation across the jet at the nozzle outlet, as given inFigure 58, is replotted in Figure 60 as the ratio f.ior comparison withtheory. In this display the unitized jet width is taken as 2-1/8 inches, thisbeing the greatest width at which a finite velocity was observed on the cushionside.
From the comparison of Figure 60, it is evident that the present theory givesa very fair approximation to the experimental observation. The theoey appearsto predict (1) a higher pressure at the cushion side of the jet than was obtainedexperimentally and (2) a lower pressure, implying a higher velocity, on the lipside of the jet. This is entirely plausible as the result of -iscous and diffusionlosses in the real case, not taken into account in the theories. Also, since themeasurements are not across a truly normal plane, some distortion of the axismust be present.
Flow Adjacent to the Annular Jet
In the attempt to determine the flow patterns beyond the boundaries of the jetproper, we made an extended traverse reaching from outside the jet throughthe jet and beyond for a distance of one Jet thickmess into the cushion. Theexperiment was exploratory in concept. We wanted to get some measurementsin proximity to the jet bmdMary in the high-speed region around the nozzle lip,to see if the seoondary flow inside the cushion could be detected, and, as apractical objective, to see if the variation in height of the cushion cavity wouldinfluence the performance. Accordingly, the measurements were made, firstly,with the cushion flush with the end of the nozzle, as in all the preceding experi-ments, and secondly, with the cushion board high In relation to the nozzle asfeatured in the Illustration, Figure 55.
To clear the nozzle and low cushion board, the traverse axis was slightlylower than in the nozzle measurements of Figure 58. The answers to thequestions are evident in the plots of the measurements in Figures 61 and 62.
110
LOSS IN TOTAL PRESSJRE = _
DYNAMIC HEAD AT THROAT 1-2000 0.5
1500 0
' 1000rz4
~ 500
04
S50 NOZZLE
2.0 1.0 0
DJSTANCE INWARD FROM NOZZLE LIP IN INCHES
Power at throat %C44=44& - 13? ft-lb/sec ft.Power at nozzl. - - 132.4 ft-lb/sec ft.Power lose 4.6 ft-lb/e ft.Potr loss o• -'Ated 5.49 ft-lb/sc ft.
Figure 59. eItributio of Power In Anmular Jet.
111
PINNES FREE-VORTEX THEORY11.011
0.8 1
'~0.6
0ý
0.4
S0.2
0 0.2 0.4 0.6 0.8 1.0
DISTANCE INWARD FROM NOZZLE LUP 7k
Figure 60. Static Pressure Variation Across a Jet -Comparison of Theoretical and ExperimentalValues.
112
*03 *LA MI LUZOMNrA
00Z0
b.4 r ~I
00
NNeI
~-
RMLY JOS2O[INEe[Mh
11
*339/ *JA Nil LLIOO'IZA
I r4P4.1
00o
.00o0
0ZLt JOp.OIN S f n
z ____1_
As in the preceding measurements, the values here plotted are computed fromthe new data without smoothing or selection.
The results show that there is a slight improvement in cushion pressure withthe large cavity. The blower was able to pull the outlet depression down to3. 12 inches of water, as against 2. 91 inches, with the small cavity (low cushionboard). The preceding test (Figure 58) gave slightly higher values: = 2.61,
= 2.94, . = 0. 892. This difference, when expressed in te,.ins of posi-tive supply pressure, appears as a higher supply pressure and Yigher cushionpressure. This is presumably due to long-term supply voltage variations tothe rig motor rather than a change in the total rig loss coefficient. The cushionpressure ratio for both tests 1A6 1)remains practically the same at
;A.
2.7128 = 0. 890 (large cavity)
d-5/. = = 0.887 (small cavity).,~p. 2.91
Thus, there is no difference between the two configurations, within the exper-imental accuracy of the tests.
Some movement of air, downward In the largep ovity sad upward in proximityto the Jet, was detected. The velocities wore In the ruagsof 0 - 10 feet persecond and were presumably evidswe of a prima"y cuslin vortex driven bythe Jet. With the low oushlon board, practically ao i movement was detected;there were traces of flow at 5 feet per seond or so toward the jet along theunderside of the cushion board bu the O realty these Is somewhat questionable.
In all other rspects the traverses are almost etWial sad Consisten with themeasurements closer to the jet (Figure 58). This oonsistemy, Incidentally,increases confidence in the aocuracy of the nstrumenatien used in these ex-periments.
115
Chapter Five
THE EFFECT OF TOTAL PRESSURE VARIATION ACROSS AN ANNULAR JET
The general solution for an annular jet (Equation 94) contains the local jet totalpressure ( A 5 ) as a variable of . So far, the equation has been solved onlyfor the case A ff = constant.
In a practical application AP, is rarely constant, because of the losses incurredupstream of the nozzle. Thus it is of interest to discover the effect of varyingAR• across the jet.
From a more academic point of view, there io no reason to suppose that con-stant total pressure gives the maximum lift per unit jet power expended, andit would be instructive to discover the true optimum distribution. in practicalcases where some adjustment of the q distribution is possible (an annularjet driven as an eductor, for example) it may then be possible to bias the distri-bution in the direction of this theoretical optimum.
There is also another practical aspect. It is usual to reduce experimentalmeasurements of cushion pressure by dividing by the mean jet total pressure.If this parameter is not independent of the total pressure distribution shape --as we shall see that it is not -- then this method of presenting results is unsat-isfactory.
THE SPECIAL CUS OF CONA1ANT JET VELOCITY
In this smction the samular jet equMtons will be solved for the special case ofan smlar jet Wbose tota bead distriutian is such that the jet velocity is con-stant across the duct.
Ap (174)
Equation (174) can be substituted forAP& In the basic equation of motion. FromEquation (97),the auatic. for ourved flow becomes
1(175)
116
or
(176)
A, 4-' .•,. ,7- 7 - (177)
1 ,I 4vi f 2 4-i Q (178)
Substituting in (174) for
P. +
.3'i( .. '¼ i0 (179)
The cushion pressure is obviously
S 870 (181)
-e, .• % 11 4- (182)
The Jet power is
117
Thus, by substituting(177) or (178) in (183), and by using equation (181) or (182)
for the cushion pressure, we can determine the parameter A+L
Determination of the Jet Curvature Parameter 2
The nozzle mass flow is obviously
,;-~ M (184)
At ambient pressure, the local velocity will be given by
e~i 'jf2. Y2.f .(15
0
Combining (184) and (185) gives the integral relationship
)~ (186)
But, from Equation (107)
Theme two equations enable us to determine .
Calculation of h4, for Free-Vortex Flow
For 7 = 1.0 Equations (180) and (183) give
:.__,•.- , 1 - . (187)
118
Thus,from Equation (145),
ej LIL(188)
Since -,t )
= .Cec.-'l (, - .*k)Ij . (189)
Calculation of for Constant Radius Flow
For = 0, Equations (179) and (183) give
C•- t-o +C, (190)
ThuA from Equation (145),
(191)
where i7# I in this case.
Discussion of Results
The total head variations necessary to give constant jet velocity are given inFigure 63 expressed as the ratio APQ/AEP.. This ratio uniquely defines Afor the constant radius of curvature case, since the variation is linear acro isthe jet (Equation 179),but not the free-vortex solution.
Values of .hAOý. plotted in Figure 64 show a substantial difference betweenthe two curvature assumptions, but little difference in each case for constant
119
TOTAL HEADDISTRIBUTION
5
CONSTANT RADIUS
4 VI5 . 1 .0/APIL
FREE-VORTEX FLOW
3
2
04 6 8 10
HEIGHT PARAMETER R/t
Figure 63. Jet Total Head Distribution Needed To Give
Constant Jet Velocity.
120
5.0__ _ _ _
5. 0 APj = CONSTANT... 4 rc _____ ____ = ONSTANT
CONSTANT RADIUS FLOWFREE-VORTEX FLOW
2.0 -
1.0
0. 5 ______ _____ _
0.2
0.1
.050 2 4 6 8 10
HEIGHT PARAMETER Alt
Figure 64. The Cushion Pressure Parameter A)i
as a Function of the Height Parameter
121
jet velocity as against constant total head, except at very low values of R .Figure 65 emphasizes this point, and it is interesting to note that, whereas con-stant jet velocity increases the lift per unit power for constant radius theory, itdecreases it in free-vortex flow. Only the constant radius solution is significantfor % 4 1. 0, of course.
THE INFLUENCE OF THE TOTAL PRESSURE DISTRIBUTION ON CUSHIONPRESSURE IN TERMS OF MEAN TOTAL PRESSURE, FOR FREE-VORTEXFLOW
It is normal to express cushion press•re as the ratio , For example,exponential theory gives
for a constant total head in the jet.
We have seen that in reducing experimental data, when A is not constantacross the jet, it is usual to find the mean value a& and Ience the ratio60,3• . It will he shown below that this is acceptable when the total pressureprofile is symmetrical about the midpoint, but not when the distribution isskewed, for the case of free-vortex flow.
For the case 1= , Equation (95) gives the cushion pressure as
6 4. 43__ f
That is, ~ 7 ) Z w4 5'(;: (192)
Thus the determination of the cushion pressure depends upon the evaluationof the two integralsI
A (the mean total pressure)(193)
and (a weighted mean).
122
2.0
S' CONSTANT RADIUS OF CURVATURE FLOW
00
0 ~1.5
Cj4 -- FREE-VORTEX FLOW
2 4 6 8 10
HEIGHT PARA]METER */
Figure 65. Comparative Effect of Constant Jet Velocityand Constant Total Head on Cushion PressurePer Unit Air Power.
123
Equation (192) can be rewritten as
- (194)
If a shape integral is defined as
A Ps 4N) j- (195)
Equation (194) becomes
Y t= [ 2~~)J +()) (196)APM
= (value for a constant + (increment duetotal pressure distribution) to the shape factor)
In the case of profiles which ai-e symmetri about -•=/ it can be shownthat ) 0 always. TU is Iprovd as follows:
S-
.. ./ ,' - • •i ,¢ !, ;
124
/OA ./) i4 Bj$ 7